Wave Propagation (C.I.M.E. Summer Schools, 81)

Giorgio Ferrarese ( E d.)

Wave Propagation Lectures given at a Summer School of the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Bressanone (Bolzano), Italy, June 8-17, 1980

C.I.M.E. Foundation c/o Dipartimento di Matematica “U. Dini” Viale Morgagni n. 67/a 50134 Firenze Italy [email protected]

ISBN 978-3-642-11064-1 e-ISBN: 978-3-642-11066-5 DOI:10.1007/978-3-642-11066-5 Springer Heidelberg Dordrecht London New York

©Springer-Verlag Berlin Heidelberg 2010 st Reprint of the 1 Ed. C.I.M.E., Ed. Liguori, Napoli & Birkhäuser 1982 With kind permission of C.I.M.E.

Printed on acid-free paper

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CON TEN T S

C

0

u r s e s

7

A. JEFFREY

Lectures on nonlinear wave propagation

Page

Y. CHOQUET-BRUHAT

Ondes asymptotiques ••••••••••••••••••

"

99

G. BOILLAT

Urti •••••••••••••••••••••••••••••••••

"

167

S e min a r s D. GRAFFI

Sulla teoria dell'ottica non-lineare

"

195

G. GRIOLI

Sulla propagazione del calore nei mezzi continui •••••••••••••••••••••••••••••

"

215

T. MANACORDA

Onde nei solidi con vincoli interni

"

231

T. RUGGERI

"Entropy principle" and main field for a non linear covariant system ••••••••

"

B. STRAUGHAN

Singular surfaces in dipolar materials and possible consequences for continuum mechanics ••••••••••••••••••••••••••••

"

275

CENTRO INTERNAZIONALE MATEMATICO ESTlVO (C.I.M.E.)

LECTURES ON NONLINEAR WAVE PROPAGATION A. JEFFREY

CIME Session on Wave Propagation Bressanone, June 1980

Department of Engineering Mathematics, The University Newcastle upon Tyne, NEl 7RU, England

9 CONTENTS Lecture 1.

Lecture 2.

Lecture 3.

Lecture 4.

Fundamental Ideas Concerning Wave Equations

1-1

1.

General Ideas

1-1

2.

The Linear Wave Equation

1-2

3.

The Cauchy Problem - Characteristic Surfaces

1-5

4.

Domain of Dependence - Energy Integral

1-9

5.

General Effect of Nonlinearity

1-13

References

1-15

Quasilinear Hyperbolic Systems, Characteristics and Riemann Invariants

2-1

1.

Characteristics

2.

Wavefronts Bounding a Constant State

2-1 2-6

3.

Riemann invariants

2-8

References

2-12

Simple Waves and the Exceptional Condition 1.

Simple Waves

3-1

2.

Generalised Simple Waves and Riemann Invariants

3-2

3.

Exceptional Condition and Genuine Nonlinearity

3-6

References

3-9

The Development of Jump Discontinuities in Nonlinear Hyperbolic Systems of Equations

4-1

1.

4-1

General Considerations

2.

The Initial Value Problem

4-2

3.

Time and Place of Breakdown of Solution

4-2 4-9

References Lecture 5.

Lecture 6.

3-1

The Gradient Catastrophe and the Breaking of Water Waves in a Channel of Arbitrarily Varying Depth and Width

5-1

1.

Basic Equations

5-1

2.

The Bernoulli Equation for the Acceleration Wave Amplitude

5-2

3.

The Amplitude a(x) and its Implications

5-3

References

5-5

Shocks and Weak Solutions

6-1

1.

Conservation Systems and Conditions Across a Shock

6-1

2.

Weak Solutions and Non-Uniquenes&

6-4

10

3. 4. Lecture 7.

Lecture 8.

Conservation Equations with a Convex Extension

6-11

Interaction of Weak Discontinuities

6-13

References

6-14

The Riemann Problem, Glimm's Scheme and Unboundedness of Solutions

7-1

1-

The Riemann Problem for a Scalar Equation

7-1

2. 3.

Riemann Problem for a System

7-3

Glimm's Method

7-5

4.

Non-Global Existence of Solutions

7-8

References

7-10

Far Fields, Solitons and Inverse Scattering

8-1

1-

Far Fields

8-1

2. 3.

Reductive Perturbation Method

8-3

Travelling Waves and Solitons

8-6

4.

Inverse Scattering

8-9

References

8-13

11

Lecture 1 . 1.

Fundamental Ideas Concerning Wave Equations

General Ideas The physical concept of a wave is a very general one.

It includes the

cases of a clearly identifiable disturbance, that may either be localised or non-localised, and which propagates in space with increasing time, a timedependent disturbance throughout space that mayor may not be repetitive in nature and which frequently has no persistent geometrical feature

that can

be said to propagate, and even periodic behaviour in space that is independent of the time.

The most important single feature that characterises a wave

when time is involved, and which separates wave-like behaviour from the mere dependence of a solution on time, is that some attribute of it can be shown to propagate in space at a finite speed . In time dependent

situat~ons,

the partial differential equations most

closely associated with wave propagation are of hyperbolic type, and they may be either linear or nonlinear.

However, when parabolic equations are

considered whicp have nonlinear terms , then they also can often be regarded as describing wave propagation in the above-mentioned general sense.

Their

role in the study of nonlinear wave propagation is becoming increasingly important , and knowledge of the properties of their solutions , both qualitative and quantitative, is of considerable value when applications to physical problems are to be made.

These equations frequently arise as a result of the

determination of the asymptotic behaviour of a complicated system. Nonlinearity in waves manifests itself in a variety of ways, and in the case of waves governed by hyperbolic equations, perhaps the most striking is the evolution of discontinuous solutions from arbitrarily well behaved initial data .

In the case of parabolic equations the effect of nonlinearity

is tempered by the effects of dissipation and dispersion that might also be present.

Roughly speaking, when the dispersion effect is weak, long wave

behaviour is possible, whereas when it is strong a highly oscillatory behaviour occurs, though the envelope of the oscillations then exhibits some of the characteristics of long waves.

12

Waves governed by a linear wave equation arise in many familiar physical situations, like electromagnetic theory, vibrations in linear elastic solids, acoustics and in irrotational inviscid l i qui ds .

However,

these linear equations often arise as a consequence of an approximation involving small amplitude waves, so that when this assumption is violated the equations governing the motion become nonlinear. Not only does this convert the problem to one involving nonlinear partial differential equations, but it also usually leads to the study of a system of first order equations, rather than to a nonlinear form of the familiar second order wave equation.

This happens because the wave equation

usually arises as the result of the elimination of certain dependent variables from f irst order equations (like! or

~

in electromagnetic theory),

and this is often impossible when nonlinearity arises .

Our concern hereafter

will thus be mainly with quasilinear first order systems of equations - that is to say with systems that are linear in their first order derivatives, and for the most part we will confine attention to one space dimension and time. 2.

The Linear Wave Equation Because of the importance of the linear wave equation (1)

let us begin by reviewing some of the basic ideas that are involved, though in the more general context of the variable coefficient equation 3

r i,j-O with a

i j,

(2)

f

b

i,

c , f functions of the four dimensional vector

~

012 (x , x ,x

Not all linear second-order equations of this form describe wave motion , and on account of this it is necessary to produce a method of classification which readily allows the identification of wave type equations from amongst the other types that are possible (i.e. elliptic and parabolic). The form of class ification to be adopted utilizes the coefficients of the highest-order n?-rlva. !:iv ee and ha s an algeb r a i c ba s i s but, as will be s een

3 x).

13 in a subsequent section, this classification in fact effectively distinguishes between equations of wave type and those of other types.

Let us start by

attempting some simplification of the form of equation (2) by changing the independent variables through the linear transformation i

0,1,2,3

(3)

where the cl(ij are constants. A transformation of this form gives an affine mapping of the (xO, xl, x

2,

x 3)-space which is one-one provided

detl~jl; 0.

Employing

the chain rule for differentiation we find that equation (2) may be re-written 3

°.

}:.

i,j,k,R.=O Hence the coefficients a

(4)

of the derivatives u i j' which are functions of x x position, transform to the new coefficients ij

of the derivatives u k t' which are also functions of position.

If, now,

; ;

we confine attention to the set of coefficients a specific point

~

""

~

appropriate to some ij 012 3 in (x , x , x , x )-space, we see that this is exactly

the transformation rule which would apply to the coefficients a

ij

of the

quadratic form 3

L

i,j=O

aijTliTlj ,

(5)

when the Tl i are transformed to 8 by the variable change k 3 Tli a ki8k• k-O

r

Now it is a standard algebraic result that by a suitable transformation a quadratic form with constant coefficients may always be reduced to a sum of squares, though not all of the squared terms need be of the same sign. Furthermore, Sylvester's law of inertia asserts that however this reduction is accomplished , the number of positive terms m and the number of negative

14 terms n will always be the same.

To apply these results to the differential

equation (2) itself with the variable coefficients a attention to a fixed point

~

let us again confine

i j,

o 1 2 3 • !o in (x , x , x , x )-space and attribute to

the a i j the specific values a i j - aij(!o). This then implies that some choice of the numbers a

ij

•

ai j

exists for

which

where m + n < 4.

The number pair (m,n) is called the Signature of the

quadratic form (5) and, being an algebraic invariant, is used to classify the quadratic form.

We shall use it to classify the variable coefficient

partial differential equation (2) at each point

~

= !o'

The effect on equation (2) of using these numbers

ai j

in the transformation

(3) is to yield at ~ - !o a differential equation of the form

m-l

I i=O

u

~i~i

-

m+n-l

3

I

u i i +

i"'1ll

~ ~

I

i:oO

biu i + f t

0

(6)

Equation (6) or, equivalently, (2) is called hyPerbolic at ~ = !o in the o ~ -direction when the signature is (1,3), elliptic when the signature is (4,0) and parabolic when m + n < 4. direction at each point of a region the

o ~ -direction

throughout

If an equation is hyperbolic in the ~O_

n, then it is said to be hyperbolic in

n.

Obviously, if an equation has constant coefficients, then one suitable transformation (3) will reduce it to the form of equation (6) throughout all space.

For example, aside from the trivial transformation to remove the

constant factor I/c 2, the wave equation (1) is already seen to have the signature (1,3).

Thus if a transformation is made at one point of space to

convert the factor llc

2

to unity, then it does so for all points in the space.

The usual effect of variable coefficients and first-order terms in hyperbolic equations of the form (2) is to introduce distortion as the wave profile propagates.

This produces various complications, not die least of

which is the fact that the wave velocity becomes ambiguous and requires

15 careful definition.

Only when there is a clearly identifiable feature of

the wave which is preserved throughout propagation is it possible to define the propagation speed of this feature unambiguously.

Such is the case with

a wave front separating, say, a disturbed and an undisturbed region and across which a derivative of the solution is discontinuous. 3.

The Cauchy Problem - Characteristic Surfaces Fundamental , to the study of hyperbolic equations is the Cauchy problem,

and the associated notion of a characteristic surface.

In brief, when

working with four independent variables the Cauchy problem amounts to the ~etermination

of a unique solution to an initial value problem in which a

hypersurface F is given, and on it the function u is specified together with the derivative of u along some vector directed out of F. directional derivative is

call~d

Such a

an exterior derivative of u with respect

to F, in order to distinguish it from a directional derivative in F which is known as an interior derivative.

In the Cauchy problem it must be

emphasized that the function u and its exterior derivative over the initial hypersurface F are independent, and can be specified arbitrarily. A hypersurface F for which the Cauchy problem is not meaningful because u and its exterior derivative cannot be specified independently is called a characteristic hypersurface.

Let us now see how characteristic hypersurfaces

may be determined. It is convenient to utilize curvi-linear coordinates

~

012 , ~ , ~ ,

~

3

and

to let the hypersurface F on which the initial data is to be specified have the equation

~

o = O.

In terms of the new variables, a derivative with respect

to ~O is a directional derivative normal to F so that it is an exterior derivative, whilst derivatives with respect to

123 , ~ , ~ are interior

~

derivatives. We now utilize this by rewriting equation (2) in a form in which the derivative u

~o~O

is separated from the other second-order derivatives

16

3

+ Here

,

L

L

(7)

f •

i,kwO

signifies that the terms corresponding to k

= i

=0

are omitted from

the summation. Now if we specify u and u 0 independently on F, as is required in the E;

Cauchy problem, the substitution of their functional forms into equation (7) enables the determination of u 0 0' provided only that the coefficient E;

of this derivative does not vanish.

E;

Thereafter, the solution may be obtained

in the form of a Taylor series by determining the coefficients of the series by successive differentiation of the initial data and of equation (7) itself. This is, of course, the idea underlying the classical Cauchy-Kowalewski theorem.

It is, however, very restrictive as an existence theorem since

..

it demands that all functions involved are C • In the event that the coefficient (8)

of u 0 0 vanishes, neither this nor higher-order derivatives of u with E; E; 0 respect to E; can be found . Furthermore, the derivative u 0 0 may then be E; E;

specified arbitrarily on F, and even differently on opposite sides of F. This is not remarkable, because when the coefficient of u 0 0 vanishes, u and u 0 cannot be specified independently over F.

E;

E;

This follows because

E;

they must satisfy the equation which results when the first term is deleted from equation (7), and so we then have insufficient initial data.

As already mentioned, the hypersurface F with the equation E;0

=0

for

which the coefficient (8) vanishes is called a characteristic hypersurface of the differential equation (2). begin by setting Pi H

=

=

3E;

3

r

i,j=O

ai,Pi Pj ' J

olaxi

To examine such hypersurfaces further, we and writing (9)

17 Then the quadratic form H is the coefficient of the derivative u 0 0 in ~ ~

equation (7), and the characteristic hypersurface F will be given by the condition H

= O.

To interpret the condition H

= 0,

differentiable scalar function, then

~ = const.

we first recall that if

grad~

Consequently, by analogy , Pi

~

is a

is a vector normal to the surface

= a~Olaxi

is the ith component of the

four-dimensional gradient of ~O and so is the ith component of a four-dimensional vector

~

normal to the hypersurface F.

Hence the equation H • 0 is a

condition on the orientation of the normal vector

~

to F, and as the a

ij

are

usually functions of position, it follows that this condition will differ from point to point. The quadratic form (9) is, of course, just the same quadratic form we encountered in (5), so that its signature will depend on the type of the equation (7) or, equivalently, (2). ~ =

!o

If the equation is hyperbolic at

the signature will be (1,3), and it follows that at the point the

condition H

=0

determining the characteristic hypersurface can be reduced to (10)

It is obvious that no real characteristic hypersurface exists for elliptic equations, since their signature is (4,0) and the components of the vector

~

need to be complex if they are to satisfy the condition

222 2 H • Po + PI + P2 + P3

0 •

To proceed with the hyperbolic case we now simplify matters by setting O x • t and writing ~O

t - ~(xl, x 2 , x 3)

so that Po • 1 and Pi becomes ~2 + ~2 2 x1 x

+ ~2 3 x

= -~

x

1

i for i

(11)

= 1 ,2,3.

Then the quadratic form (10)

(12)

18

which is a differential equation for the function

~

locally at

~ = ~.

This is, of course, the familiar Eikonal equation from mathematical physics . At any time t

1

~(x ,

=

2

to a real three-dimensional surface S is defined by

3

(13)

x , x )

and this is called a characteristic surface . If equation (7) is a constant coefficient equation it can be reduced to the form of equation (6) with m

= 1,

n

=

3 throughout all space, so that

equation (12) then describes the characteristic surface

= const

~

for all

points in space . In summary, we have established that real characteristic surfaces

oc~ur

in connection with hyperbolic equations, and that across such surfaces a discontinuity may occur in the second normal derivative of the solution.

This

discontinuity in a derivative of a solution is usually identifiable with an interesting physical attribute of the solution, since it represents a wavefront bounding two regions. The discontinuity surface, or wavefront, advances with time, as is shown by the following simple argument. Taking the total differential of ~o

123

dt - dx • 1 - dx • 2 - dx • 3 x x x

=0

and using equation (11) we find

0

or , equivalently dt

= .d£ . gra d.

,

where d£ is the vector differential with components (dx

l,

2 3 dx , dx ) .

Hence 1

(14)

v.n

Igrad" where

Y..

=

dr dt

grad.

Igrad' I

The vector n is the unit normal to the surface •

s

the displacement of a position vector with time, y..

const, and as d£ represents

= d~/dt

is the velocity of

19 displacement of a specific point on the surface as the characteristic surface moves from its position at time t to its position at t + dt .

The

scalar v.n is the normal velocity of propagation of the characteristic surface or wavefront and, in general, is a function of position. By re-writing equation (7) and differencing it across the characteristic surface , we shall see that there may also be a discontinuity in the first normal derivative of the solution and this, like the discontinuity in the second-order derivative, is propagated with the characteristic surface. The equation governing the development of the discontinuities in first- and second-order derivatives is an ordinary differential equation defined along a curve in space and is called the transport equation. 4.

Domain of Dependence - Energy I~tegral The dependence of a wave solution on initial data is most easily

illustrated in terms of the one-dimensional wave equation (15)

with the initial conditions u(x,O)

hex)

au at

and

k(x) •

(x,O)

(16)

The explicit d'Alembert solution h(x-ct)+h(x+ct) 2

u(x,t)

shows how the solution at (xo,t xo - cto

~

x

~

xo + ct

+ J:.... 2c O)

x+c t k(s) ds x-ct

f

(17)

depends only on data in the interval

o

This is called the domain of dependence of the solution at (xo,t

O)'

This

same idea generalises to quasilinear hyperbolic systems and we shall employ it later. In conclusion, to illustrate the important notion of an energy integral that arises when working with equations derived from the conservation of physical quantities, let us prove the uniquenes s of the solution to the

20

?Co

Domain of dependence Cauchy problem for slightly generalised two dimensional wave equation au q(x,y)u - rat u

l

(18)

(x,y) ,

and where we shall assume P, k, r to be positive constants and q(x,y)

(19) >

O.

It will be convenient to consider that (18) governs the motion of a membrane with density P, tension k per unit length, distributed springing under the membrane with spring constant q(x,y) per unit area and fyictional coefficient

r. Then the potential energy within a fixed region R with boundary B of the (x,y)-plane comprises the energy stored in the springing qu

2

dxdy

and the energy stored in the membrane

- .! 2

+.! 2

II I

2u 2 Uk[a 2 + a R ax

1B

alu)

uk au

ax

dxdy

ds

with n the outward dfuwn unit normal to Band ds a length element of B.

The

first integral in ~(t) is the negative of the work done by the tension against the interior of R and the second integral the negative of the work done against the boundary. Green's theorem shows that

so that the total potential energy

21

'; 0 • The transformation introduced through equations (10) is non-singular provided the Jacobian •

1 ~x

(14)

is non-zero and finite.

The initial condition on

~

ensures that

x~

is initially equal to unity and

so we may assume the non-vanishing of the Jacobian for at least a finite time

= O.

after t

Let us denote by L the open region lying to the left of the

advancing wavefront (x,t) • 0 and bounded on the left by the characteristic (x,t)

=

0 also issuing out of the origin and chosen so that no other

characteristics enter L.

Then, since no characteristics enter L, U will

remain smooth in L for at least a finite time. operations on the side

~

< 0

All subsequent limiting

of the wavefront will be assumed to be performed

in L. Let l(j) be the left eigenvector of A corresponding to the eigenvalue A(j) then, from equation (3)

o . Employing the identities

...!. =!t ...!.. + l!'...!.. at - at

a~

at at'

and

...!. =!t...!.. + l!•...!.. ax

- ax

we see that

a~

ax at'

o

$1 Thus, using these results and equation (lOb) together with condition (14) to ensure the non-vanishing of the Jacobian, we obtain R. (j)

(x41 ...!. at'

+ (>. (j) - >. w)

...!.) a41

U

+

k

-

bo(j)

o.

x 41

(15)

In particular, if >.(j) - >.(41), we have ,(4I,k) + b(4I,k) Ut' ..

o ,

1,2, ••• ,r

(16)

41

is the multiplicity of the eigenvalue >.(41). 41 By virtue of our choice of coordinates the wavefront 41 - 0 is a

where r

characteristic, and since the solution is Lipschitz wavefront, jump

con~nuous

across the

discontinuities in derivatives with respect to 41 may take

place across 41 - O.

Accordingly we define the jumps across 41 - 0 as follows:

41-0- 41-0+

U is continuous :

[U ]

Ut' is continuous

[Ut']4I-O+

41=0-

-

0

or

U(O,t') - Uo

(constant)

0

U is discontinuous 4I and X

41

is discontinuous

We note that since both IT and X depend on 41 they are not independent and we shall later determine their precise relationship [see equation (21)]. From the definition of X we see that X + (x

- (x4')4I_0_,whi1e 4l)4I=0+ Hence, in a neighbourhood of the wavefront

(x4')4I-O+ - x 0 (say) is finite. 41 condition (14) is seen to be equivalent to the condition

x+

x 0 is finite and non-zero. 41 The significance of the non-vanishing of the Jacobian may easily be

(14')

seen by noting that in L and along 41 - 0 we have U

x

whence

So, if x

41

vanishes while U4' remains finite, U ceases to be Lipschitz continuous

and we have the gradient catastrophe.

52 In the simple case that U

o

m

constant the jump conditions on n and X

reduce to n(t' )

and X(t') •

(17)

So, since t(j) is assumed to be continuous across the wavefront and Ut' is continuous across the wavefront with U , t

=0

in the constant region we

have, by using (9) and by considering equation (15) at a point P in Land letting the point tend to a point on the wavefront, that j

(18)

r.p+l' ••• , n

where aga in the subscript 0 signifies the constant state appropriate to (9). We now differentiate equation (16) with respect to .p at point P in L to obtain

or,

where T denotes the. transpose operation and where V is the gradient operator u with respect to (u

l'

u

2,

•••• uu)-space.

Again, letting P tend to a point

on the wavefront and using the fact that U is continuous across the wavefront t, with (Ut').p_O+= 0 we obtain the equation k

m

1,2, •• • ,r.p

(19)

If, now, we differentiate equation (13') with respect to .p at a point P in L we obtain 0

~

(~~, )

(V A(,» u

U,

and so, 0

at' (x,)

(V A(.p»U u ,

(20)

53 Thus, again letting P lend to a point on the wavefront, and using (17) we find that (21)

i~j) are linearly independent vectors we may use equations

Since the (18) to express TI.

(n-r~)

components of TI in terms of a certain

components of

r~

Introducing these expressions into equations (19) leads to

r~

first order

ordinary differential equations with constant coefficients for the say TIl' TI

. •. , rrr~ .

2,

r~

unknowns,

Introducing TI thus determined into equation (21) and

integrating the result we may obtain an expression for X.

However, before

doing this it is necessary to define certain limiting operations that will be necessary in the integration process. ~

If Q is a quantity defjned only in L we define the operation Q to be the limit

Q_ lim

t' ->()

(Q)~~o-

adjacent sides of ~ taken along

=0

For a jump quantity P depending on the state On we define the operation

P to

be the limit

o.

'~ ~

P = lim

t' ->()

P

Thus, integrating equation (21) with respect to t' between 0 and T and noting that X is a jump quantity defined across operation

x ..

X just

~

.. 0 we may use the limiting

defined to obtain the result

x+ ITo (V A(~) u

TIdt' •

(22)

0

This equation describes the variation of X along the wavefront

~

.. 0 with

advancing time and in writing equation (22) we have tacitly assumed that multiplicity

r~ of A(~) remains unchanged .

and at t' .. Tl(T

l

the

Should this assumption not be true

< T) the multiplicity changes, then TI must be re-determined

for the interval t' > T l•

We remark here that although an initial condition

on Ux may be prescribed arbitrarily by specifying lim (Ux)t-O' this limiting x->() operation is not in L and so in general

Ux is

not equal to this limit and

although on the initial line may not be prescribed arbitrarily. Equation (22) may be displayed in a slightly different form from which the critical time t

c

may be determined as follows.

By definition

54 it = or

when equation (22) becomes

However, from (14') we see that the left-hand side of -t hi s expression is simply the Jacobian of the transformation and so is required to be finite and non-zero in order that the transformation is unique. critical time Jacobian x$

T

= tc

=X+

x 0 $

So, if there is a

at which condition (14') ceases to be valid and the -

0, this is given by

(23) Geometrically the vanishing of the Jacobian is equivalent to the point at which the $ - constant lines first intersect the wavefront $

= O.

(i.e ., the

family of characteristics $ - constant intersect at a cusp) . To determine t

c

in terms of U we divide equation (23) by x$ to obtain x

o. The vector

n must

be determined from equations (18), (19).

(24)

In the particularly

simple case that B • 0 it follows directly from (18) and (19), provided the multiplicity of A($) is constant, that n is a constant equal to its initial value

.

n,

and so

n • and thus

ux Using the definitions of X and n we see that

55

when x

4

..

but

and so U is given in L by the expression x

UIII + X

U

X

(V

u

~(4»

0

Ut'}. x

(25)

Thus Ux becomes unbounded if the denominator vanishes for some t c > O.

If

(Vu~(4»o = 0 it follows directly from equation (21) that X is a constant and so t

c

is infinite.

The discontinuity in this case is propagated but

remains finite for all time.

Systems for which this property is true are

a special case of those which are exceptional with respect to the

~(4)

characteristic field. The general case when A, B depend on U and also explicitly on x, t has been discussed in detail in [1].

A different approach to the problem

that involves three space dimensions and time has been described by Boillat [2] and Chen [3). References [1] [2] [3]

A. Jeffrey, Quasilinear Hyperbolic Systems and Waves. Research Note in Mathematics No .5, Pitman Publishing, London, 1976. G. Boillat, La Propagation des Ondes. Gauthier-Villars, Paris,1965. P. J. Chen, Selected Topics in Wave Prop~gation. Noordhoff, Leyden, 1974.

50

Lecture 5. The Gradient Catastrophe and the Breaking of Water Waves in a Channel of Arbitrarily Varying Depth and Width

1.

Basic Equations To illustrate the gradient catastrophe in a physical context, let us

show how to obtain an explicit form for the amplitude of an acceleration wave that propagates into water at rest which is contained in a vertical walled channel with slowly varying width W(x) and an arbitrarily varying depth hex) below the equilibrium water level.

The method we describe is

taken from the joint paper submitted for publication to ZAMP by the author and J. Mvungi [1]. As usual, let the x-axis lie in the equilibrium surface of the water

in the direction of propagation, with the y-axis pointing vertically

= o.

upwards, and write the equation of the bottom of the channel as y + hex)

Then, if the elevation of the water above the equilibrium level is n(x,t), g is the acceleration due to gravity and the x-component of the water velocity is u(x,t), the equation of motion in the x-direction is as derived by Stoker

[2], namely (1)

However, the equation corresponding to the conservation of mass will now be different on account of the width variation of the channeL

To derive

it, all that is necessary is to observe that the cross-sectional area S(x,t) of the water at any given place and time (x,t) is S(x,t) and that the flow through this area is S(x,t)u(x,t).

= W(x)(n(x,t)

+

hex»~,

Thus, equating the

time rate of change of S to the negative flux through it, we find

-(Su)

x

(2)

,

from which it follows that

o

(3)

The governing equations for flow in a variable width channel of arbitrary depth are thus equations (1) and (3).

The assumption of a slow

variation in the width is necessary because the transverse movement of the

57 water has been neglected in these one-dimensional long wave equations, . and this will cease to be a good approximation if the width changes too rapidly. 2.

The Bernoulli Equation For The Acceleration Wave Amplitude Suppose the wave moves in the direction of increasing x, starting

from x • 0 at t • 0, and that it moves into water at rest.

Then, across

the wavefront: (i)

u and n are continuous, with u(x,t) • n(x,t) • 0 ahead of the advancing wave,

(ii)

the first and second derivatives of u and n suffer at most a jump discontinuity, so that the wavefront being propagated on the surface is an acceleration wave.

Using a superscript minua sign to denote the value of a function immediately behind the advancing wavefront (i.e . at the edge of the disturbp.d region) we conclude from (i) that u

n

o

(4)

Taking the total differential of equations (4) gives, just behind the wavefront, n-dx + n-dt x t

~+ u~dt and 0

-0

or, equivalently, u

t

-cux

and

nt

-c n

(5)

x

where c • dx/dt is the speed of propagation of the wavefront which is, of course, a characteristic curve for the system (1), (3) . Immediately behind the wavefront (1) and (3) become

o

and

n~ + hu; -

(6)

0,

where it is understood that h • hex) is the depth at the wavefront. n; ; 0 equations (5) and (6) imply the standard result c

2

If

• gh.

Now define the amplitude of the acceleration wave to be

a

a(x)

• nx '

(7)

58 when (5) and (6) become

"e- ,. -ga

and

ux •

(8)

ga/c.

Now notice that the operation of differentiation with respect to x along the characteristic followed by the wavefront, behind which u~ and u~ are defined, takes the form (9)

It then follows immediately from this that •

c2

U

xx

-

U

(10)

tt

To obtain the differential equation governing the behaviour of the amplitude a of the acceleration wave we first differentiate (1) partially with respect to t and (3) partially with respect to x . n

xt

Then eliminating

' and using (7) and (8), we find gcW_ 3 2 c 2u- - u- + (2lh __ x + _ x ) a +.=.lL xx tt c W c

a2

,.

o.

(11)

Combining (10) and (11) and using (8) brings us to the required Bernoulli type equation for the amplitude a(x), W) 3a2 da + 3h (4h x + 2; a + 2h dx

,. o ,

(12)

in which use has been made of the fact that, as c 2

gh, we have (dc/dx)

=

gh/2c. 3.

The Amplitude a(x) And Its Implications The standard substitution a • b

-1

reduces the Bernoulli equation (12)

to a linear first order equation, and a simple calculation then shows that (13)

in which a

I(x)

O

o•

• a(O), W

3h 3/4w 1/2 o 0 2

W(O) and

(14)

59 A wave of elevation corresponds to a to

"o

O

< 0 , and a wave of depression

The wave will be said to break if for some x

> O.

g

x

c

the water

surface behind the wavefront becomes vertical, so that the amplitude a(x (a)

g

c)

Since lex)

m.

~

0, we conclude from the form of (13) that:

A wave of elevation (a . < 0) in a variable width channel always

o

breaks in water of finite depth provided lex) is such that 1 + aOI(x 0, and Xc > 0 is finite.

at x

~

c)

g

If the depth of the water shelves to zero

t, say, so that h(t) • 0, a wave of elevation propagating

towards the shore will break before reaching the shore line if laol > l/l(t), and at the shore line if laol ~ let). (b)

A wave of depression (a

O

> 0) in a variable width channel can only

break if the depth of the water shelves to zero, and then only at the shore line provided let) < When we set W(x)

= WO'

m.

these general conclusions agree with the

special case of waves climbing a beach that was studied by Greenspan [3]. This is because the one-dimensivnal long wave equations do not distinguish between flow in a parallel channel and unrestricted one-dimensional flow. Result (14) shows that the integrand of importance in this case combines the depth function hex) and the width function W(x) in the form (h(x»-7/4(W(x»-1/2.

Thus any modification of the depth and width that

leaves this combination invariant will lead to the same conditions for breaking provided a O' hO and Wo are unchanged. As special cases of results (13) and (14) we observe first that in

a parallel channel of constant depth h we obtain Stoker's result [2], that breaking occurs when x

c

2h - 3a O

at a time t

c

2 - 3a

O

(E.g) 1/2

(

Secondly, when hex) • h - mx so that the bottom has a constant slope, we obtain Jeffrey's result [4] that breaking occurs when

(15)

60

x

c

(16) Result (16) also shows that when the water deepens at a constant rate (m




° °

so that in (13) we have F(u) • u 3/ 3. Then, &s the equation

~s

homogeneous, when it is differentiable a

non-constant solution u will be a function of x/t, and it is easily verified that the function

o u(x, t)

{

(x/t);

I

is a

cl solution

° for ° ~ x/t ~ I

for x/t


1

subject to the initial condition.

This solution is continuous

everywhere for t > 0. and it is differentiable everywhere except along each of the lines x •

° and

x - t on which, due to the continuity of u, the

generalised Rankine-Hugoniot condition holds in a degenerate form.

It is a

simple matter to verify directly that this piecewise Cl classical solution is also a weak solution.

The form of this solution 1s shown in the Figure.

This

is simply a centred rarefaction wave of the type mentioned in Lecture 3. ~

The continuous piecewise C1 solution resolving a discontinuous initial condition.

70

Another weak solution follows by observing that a discontinuous function that is a Cl solution away from the lines of discontinuity will be a weak solution provided the discontinuity condition.

L~t

us seek an even simpler

{:

u(x.t)

~atisfies

fer x/t

< k

for x/t

>

w~ak

the generalised Rankine-Hugoniot

solution of the form

k •

by choosing k to satisfy the generalised Rankine-Hugoniot condition. into (2l) coupled with the fact that the speed of shock

SubB~itution

propagation h .. k then g1ves k(l - 0)

(~-

0)

or k

..

1/3 .

The second weak solution is thus for x/t < 1/3 u(x. e) for x/t > 1/3 • and this weak solution is piecewise constant. but is discontinous across the line 3x .. t. In physical problems only one solution is permissible. so that if the

class of weak solutions is considered some selection principle must be devised to choose a unique weak solution with the appropriate physical properties.

This is usually achieved on the basis of the stability of the

solution and leads to selection methods known as entropy conditions.

This

name derives from the gas dynamic case in which both compression and rar~faction

shocks are

shock is physically ~es

~thematically

r~alisable

since it is only in that case that the entropy

not decrease ,across the shock.

~ed

possible. though only the compression

In the example just examined the

rarefaction wave is the physical solution since the shock wave is

nat. .sub.1e.

71 Some account of entropy conditions and of the associated literature is to be found in the work of Lax [1], Jeffrey [2] and in the paper by Dafermos [3].

3.

Conservation Equations with A Convex Extension When the conservation system involved is symmetric hyperbolic, the

ideas of Section 2 may be pursued in some detail without giving rise to undue difficulty.

This we do now. basing our approach on the paper by

Friedrichs and Lax [4]. Consider a system of conservation equations

3U + 3G • 3t

3x

0

(22)

•

with U and G • G(U) each n x 1 vectors and integrate it over an arbitrarily large interval [-a,a] of the x-axis .

Integrating the second term by parts

then gives rise to the equation

r -a

au at

dx

+ GI

- GI

a-a

• o.

Now for the class of solution vectors U that vanish sufficiently rapidly for large Ixl. so that G(± a,t)

+

0 as a + m, we see from the above result

and the degenerate form of Theorem I that

showing that the integral

is a conserved quantity because it is independent of t. The problem we now consider is, when is a new conservation system

av + aK • 0

at

ax

(23)

•

with V, K functions of U, a direct consequence of the original law (22). To resolve this we need

to

make a direct comparison between (22) and (23)

lib that fittlt we perform the indicated differentiations, wben these equations becOllle, rlaSp,ect:1ve1y.;

72

au + at

(V G) U

au .. ax

0

(24)

and (V V)

U

au +

at

(V K)

U

~ ax

..

0

(25)

0

Employing the summation convent ion, the j-th component of ( 24) may be written

~ +~

aUt aUt ax

at

..

o ,

(26)

while equation (25) itself becomes

1Y... .-:.J + ~ ~ .. aU

j

at

aUt ax

0

(27) 0

Consequently, comparing (26) and (2 7), we conclude th at (25) will be a consequence of (26) only if (28) Let us now assume that this cond it io n is true , and dif fer ent i ate i t with re s pec t to

~,

~ a aUt [ a~

when we find 2g o

( aU av ) ) + av aUj (a j

~)

The second term on the left hand s i de and the right hand side are both symmetric in t and h, so that the f irst term must al s o be s ymmetr ic.

~e

have thus shown that if (28) is true , then

~ fa~ [~:J) . ~ [a: [~:J)

(29)

t

If, now, we multiply (26) by a

~ ~+--..iL ~ ~Uj ~~ at aUj a~ aUt

2

v/aUja~

aUt ax

..

and sum with respe ct to j we find

o .

This will be equivalent to (22) if the matrix {a

(30) 2

v/ aUj a~}

is non-singular,

and we here take note of the fact that system (30) is symmetric.

Hence,

whenever (22) is hyperbolic, and (28) is true, the equivalent system (30) will be

symmetric hyperbolic.

It can be shown that initial value problems

73 for symmetric hyperbolic equations are unique and will exist in some neighbourhood of the initial data. As the hyperbolicity of (30) implies that the matrix {a

2

v/aUja~}

is

positive definite we may assert that V is a convex function of the elements ~

and so arrive at the following conclusion.

Theorem 4 (Uniqueness Theorem) If the system of conservation equations (22) is such that it implies a new conservation equation (23) with the property that the new conserved quantity V is a convex function of the original elements u l' u

z' ... ,

un

of U, then the initial value problem for (2Z) has a unique solution in the neighbourhood 4.

ot

toe initial time.

Interaction of Weak Di,continuities We conclude this lecture by adding a few remarks about the interaction

ot ~

weak

discont inui~y

propagated along a cha r a c t e r i s t i c

c(~)

and a shock.

Here we use the term weak to refer to a solut ion which is continuous across

C(~) though its normal derivative i s di scontinuous.

This is in contrast to

the strong discontinuity pf the Rankine-Hugoniot type wher e the solution itself is discontinuous.

This situation is illustrated i n the Figure where D is the

shock line corresponding to a conservative system of the form U + A(U)U + B(U) t x /;

~

0

v.. o In general at

~where C(~) meets D, there will lie r characteristics of

the system to the right of D entering the state U+ and s to the left reflected back into

~he

state

:U~

•

74 By writing down the transport equation for the incident weak discontinuity along C(~), as in Lecture 4.it is possible to determine its nature as it approaches P from the left . 5

Then. using the fact that the r transmitted and

reflected weak discontinuities must propagate along characteristics. it

is possible to resolve the jumps across all characteristics at P in terms of the original system of equations. the differentiated Rankine-Hugoniot equation across D at P and the initial discontinuities propagating along the r + s characteristics together with C(,). Provided the set of equations that results at P is properly determined the reflected and transmitted weak discontinuities may be determined. Special cases arise. like the coincidence of D with a characteristic on either side at P, and the fact that the system may be exceptional with respect to one or more characteristic fields. A general accoUDt of these ideas is to be fouod in Jeffrey [1]. while attention was drawn by Bolllat and Ruggeri [5] to thp necessity to perturb the shock speed in cases where the interface D can move. References [1] [2] [3]

[4] [5]

Lax. P. D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM Regional Conference Series in Applied Mathematics. 11. 1973. Jeffrey. A. Quasilinear Hyperbolic Systems and Waves. Research Note in Mathematics. 5. Pitman Publishing. London. 1976. Dafermos. C. H. Characteristics in Hyperbolic Conservation Laws. A Study of the Structure and the Asymptotic Behaviour of Solutions in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium. Vol. 1. Research Note in Mathematics, 17. Pitman Publishing. London. 1977. Friedrichs. K. 0., Lax. P. D. Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci •• USA 68 (1971). 1686-1688. Boillat. G., Ruggeri. T. Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks. Proc. Roy. Soc. Edin. 83A (1979). 17-24.

75 Lecture 7.

The Riemann Problem, Glimm's Scheme and Unboundedness of

Solutions

1.

The

Riema~~

Problem for a Scalar Equation

To illustrate ideas we consider the single equation for the scalar u already encountered in connection with weak solutions in Lecture 6, namely:

o

or, equivalently,

0, (1)

The Riemann problem for this equation is then the resolution of the discontinuous initial data u(x, 0) where

o and

for

x

for

x > 0

< 0

u

are two arLitrary constants. More generally, it may be l extended to include a number of such discontinuities located along the U

initial line. The characteristics of (1) are the curves (2)

along which the equation may be written in tne form du dt

o•

K

Hence for x

(3)


ui the two families of

characteristics intersect from the start, leading to non-uniqueness and shock formation of the type first indicated at the end of Lecture 1,

76

f:'

Is~k /i.,e.

I

!I %

(b)

(a)

Centred simple wave in W

Shock speed

we thus arrive at the result that the condition for a physically admissible shock solution lor (1) is

(4) Now (1) is invariant under the replacement of x and t by aX and at, 50

that its solution depends only on the ratio t • x/to

all pass through the origin, and along them u • const.

The lines t • const. They thus fill in

the wedge shaped region Win (a), and as they are characteristics the wave solution described by them in W is called a centred simple wave.

In this

case the centre is at 0 which is the location of the discontinuity in the initial data . Taking the particular case

o•

• 1 and setting u(x, t) • u(t) l in (1) leads to the differentiable solution for region W given in U

Lecture 6, and illustrated there by 0

u(x, t)

for

(x / t ) I { l

for

x/t


1

Notice that the non-physical shock

(5)

(weak solution) given in Lec!:ure 6.

namely u (x, c)

{:

for x/t


1/3 ,

(6)

77

lies in region Wand so is

~

produced by the intersection of characteristics.

It is for this reason that it is not physically realisable and so must be rej ected. A physical shock occurs in the situation illustrated in Figure (b) however and emanates from the origin. u(x, 0)

for

It < 0

for

x > 0

Using the initial data

as a typical example, we find from the Rankine-Hugoniot condition that

A = 1/3. Thus in this case the resolution of the initial discontinuity merely involves its propagation along the shock line t • 3x. We conclude from this that for a centred simple wave (rarefaction fan) to occur, the characteristics must diverge from a point, leaving a wedge shaped region to be filled by the centred simple wave.

A shock will only

occur when the characteristics converge and intersect. 2.

Riemann Problem for a System Let us now consider the reducible hyperbolic system

subject to the initial data U(x, 0)

{::

for

x

< 0

for

x

>

0 ,

where Uo and Un are constant n element vectors.

(8)

The Riemann problem now

becomes the resolution of the initial vector discontinuity at x • 0, though as with the scalar case it may be extended to include a number of such discontinuities along the initial line. We look for the solution of this problem in terms of generalized simple waves and shocks , which will be the analogue of the situation just discussed for a single equation.

The generalized Rankine-Hugoniot

condition is of the form

A[

U])

[ F]) ,

(9)

78 once (7) has been expressed in the conservation form 3U + 3t

~ (U)

ax

•

O.

(10)

This implies n possible types of shock with speeds ~(I)C; ••• ~A(n) and we shall need a

p~1sical

we did in the simpler case.

admissibility criterion for them, just as The extension of our earlier result (4) that

provides the criterion we need is due to Lax Who requires that for some integer k with I S k

~

n

while

(11)

This condition ensures that k characteristics converge onto the shock line from the left and n - k + 1 from the right.

There is thus a

total of n + 1 conditions provided by characteristics which when taken together with the n - 1 results that follow from (9) after A has been eliminated enable the determination of the 2n values taken by U on the left ( t)

and right (r) of the shock.

The shock that satisfies (11) for some

index k is called a k-shock. Now differentiable solutions to system (7) are also expressible in terms of the ratio t • x/t, so that this system permits a generalisation of the notion of a centred simple wave.

The general solution to the

Riemann problem (7), (8) thus consists of n fans of waves, each consisting of shocks and centred simple waves, arranged in order of increasing k from left to right, and separated by sectors in Which the solution assumes constant values. As already mentioned, this generalisation of the Riemann problem may be extended to the case of an initial vector that is piecewise constant along the line t • O.

It is this very idea that is basic to

Glimm's method for the numerical solution of conservation laws (7) with arbitrary initial da ta in place of (8), and it is this that forms our next topic.

79 3.

Glimm's Method This is a method for the numerical solution of a system of hyperbolic

conservation laws

au at

+

aF (u) ax

o

(12)

with arbitrary initial data U(x, 0)

t(x) •

(13)

It is a method of first order in accuracy and the basic idea is to replace the arbitrary initial data (13) by a piecewise constant approximation in spatial intervals of length h.

Then, until such time as the

centred simple waves and shocks that result from the discontinuities interact, an analytical solution to the piecewise constant approximation to the , initial data is Riemann problem.

provi~ed

by the solution to the appropriate

If this solution is used for a suitably small time step

k, a new Riemann problem may be derived from the analytical solution at time 't .. k, and thereafter the process may be repeated to advance the solution step by step in time . The special feature in Glimm's method lies in the way in which the new Riemann problem is derived from the analytical solution.

Unlike the

averaging process over the spatial interval h used by Godunov who also employed the Riemann problem approximation, Glimm chose the constant value for each interval of length h by random sampling within that interval. Let us elaborate on this process sufficiently for its basic ideas to become clear.

First. however, we need to recall the notion of a

domain of dependence.

In Lecture 1. when discussing a second order wave

equation, the domain of dependence of a point P was defined as the interval produced on the initial line by tracing backwards in time the two characteristics passing through P until such time as they intersected the initial line. P.

Only data on this interval influenced the solution at

Now. in the case of systems (12). the analogue is to trace backwards

80 in time from a point P in the (x, t)-plane the n characteristics that are associated with system (12).

The interval on the initial line contained

between the extreme characteristics is then the domain of dependence of

P, and the Figure shows a typical example of such a situation.

o Only initial data lying within this interval can influence the solution at P . • 1

Suppose we now think of a domain of dependence of fixed length h, then for the ith interval there will be a time t is determined by the data on this interval.

i

up

to which the solution

If our initial data (13) is

approximated in a piecewise fashion at intervals of length h, then provided our time step k

t •• ;, t there will be no n} l, 2, interactions between the centred simple waves and sho cks that will result
0 for the solution v(x,t) when t. •

l/2x

(x > 0) •

This is not due in any way to the intersection of characteristics within a family, for they are parallel straight lines.

However, in this case the

initial data becomes unbounded for large x so that it might be considered this is the cause of the unboundedness of the solution.

To show this is

not the case consider this next example. Example 2 2 Take uO(x) • a tanh x, vO(x) - 1, f • l/v and g - 1, when we find

f~)

u(x,t)

[tanh(x + e) + tanh(x - t)] 2

v(x,t)

2+a[tanh(x+t)-tanh(x-t)]

Here u(x,t) remains finite for all x,t but v(x,t) becomes unbounded at an excape time t. given by t

...

tanh

-1

In this case, by making

a

suitably small, the deviation of the

initial data from constant values may be made as small as desired, but the finite escape time still persists.

84

References [1] [2] [3] [4] [5] [6] [7]

Jeffrey, A.

Quasilinear Hyperbolic Systems and Waves, Research Note in Mathematics, 5, Pitman Publishing, London, 1976. Lax, P. D. Hyperbolic Systems of Conservation Laws and the }~thematical Theory of Shock Waves, SlAM Regional Conference Series in Applied Mathematics, II, 1973. Glimm, J. Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 13 (1965), 697-715. Godunov, S. An interesting class of qua s ilinear systems, Dokl . Akad. Nauk SSSR 139 (1961), 521-523. Godunov, S. Bounds on the discrepancy of approximate solutions constructed for the equations of gas dynamics, J. Comput. Math. and Math. Phys . 1 (1961), 623-637. Chorin, A. Random choice solution of hyperbolic systems, J . Compo Phys. 22 (1976), 517-533. Jeffrey, A. The exceptional condition and unboundedness of solutions of hyperbolic systems of conservation type, Proc. Roy. Soc. Edinb. 77A (1977), 1-8.

85 Lecture 8.

Far Fields, Solitons and Inverse Scattering

Far Fields

1.

There are many different types of higher order equations and systems of equations that characterise nonlinear wave propagation in

~x

t, either

A simplification frequently takes place in

with or without dispersion.

the representation of the solutions to initial value problems to such equations after a suitable lapse of time or, equivalently, suitably far from the origin, particularly when the initial data is localised and so has compact support.

These s implified forms of solution are often asymptotic

solutions, and are appropriately called far fields. The simplest examples of these are the types of far field behaviour exhibited by the ordinary linear wave equation and by a homogeneous quasilinear hyperbolic system with n dependent variables.

Due to its

linearity, the wave equation const.)

(c

(1)

may be written either in the form (au _ (2.at + 2.) ax at c

cau) -

ax

o ,

(2)

0

(3)

or as (au + c au) (2at. _ 2.) ax at ax c

Then, if u(+) is the solution of au(±) ax

c-

o.

(4)

it follows that u(t) is a degenerate solution of (2) and u(-) is a degenerate solution of (3).

The general solution of (4) is then

f(±) (x

+ ct)

,

(5)

with f( ±) arbitrary Cl functions. Thes e travelling wave solutions are such that u(+) propagates to the right and u(-) to the left with speed c.

We t hus have the s ituation that u(±)

86 are special simple types of solution to the wave equation (1), in the sense that they only satisfy a first order partial differential equation. whereas the wave equation itself is of second order.

Such special solutions

become of considerable interest when the initial data f(±) is differentiable

o

with compact support. initial data lies in

that f

50

Ixl

(+)

O-

(x)

!

1

CO'

Then. if the support of the

d. after an elapsed time dlc the interaction


., so that

o

and

o,

(A - >'1) r O

(21)

equation (19) may be solved in the form

(22) with

~

one of the elements of U and VI an arbitrary vector function of l

The compatibility condition for (20) when solving for ilU t ilUl + t[V (V A) ] ilUl + ilT l' u 0 ~

I.

i

s-i

[_m eoa + Keoa )

~

a-I

Then taking the boundary condition U ~ U as x

o

VI : 0, we find that

~

~~.

ilt·: ilE;P

2!ilE;

is

_ 0 •

so that we may set

satisfies the nonlinear evolution equation

T.

(23)

9° c

ilP~ __ 1

•

2 ilf;P

o,

(24)

where

and

When p • 2 equation (24) becomes Burgers' equation, and when P • 3 the KdV equation.

The scalar equation (24) thus governs the far field behaviour

of the homogeneous form of system (16) that is associated with the eigenvalue A.

There will be such a far field for each real eigenvalue A of A. In concluding this section we remark that although in what follows we

shall be referring to properties of exact solutions of some far field equations, it should be remembered that these far field equations are in the main only asymptotic approximations to the solution that is of interest .

There are,

in fact , a variety of different methods by which the nonlinear evo lution equations characterising far fields may be derived, and for an account of three o ther methods we refer to the paper by Jeffrey and Kawahara , to the Scheveningen Conference paper by Jeffrey and to the AMS paper by Whitham. Before moving on to discuss soliton solutions to equations l i ke (2 4), we remark that it is a simple matter to show that the coefficient c i n fact proportional to (VuA)'r.

in (24 ) is

This means that when the characteristic

field associated with A is exceptional c vanishes.

l

l

=0

and the nonlinear term in (24)

Further analysis is required to derive the nonlinear evolution

equation that then governs the far field, and it has been shown by Jeffrey and Kakutani that a modified KdV type equation then results. 3.

Travelling Waves and Solitons We have seen that in nonlinear hyperbolic equations, waves propagate

with a shape change so that no travelling waves can exist for such

t Lons ,

That is to say, there is no reference frame moving with a constant spee1 s

91 in which the wave appears staionary.

However, when dispersion or dissipation

are present in a nonlinear evolution equation, in the sense that the linearised equation exhibits these effects as described in Lecture 1, travelling waves become possible due to the competition between the effects of non linearity and dispersion or dissipation. In

~

x t, travelling wave solutions have the form

vet) ,

v(x,t)

x - st ,

s

const. ,

and, in addition to the nonlinear evolution equation, they must satisfy some appropriate boundary conditions at infinity. determine the permissible range of values of s.

In general, these will In the case of Burgers'

equation and the KdV equation which are, respectively, examples of purely dissipative and purely dispersive nonlinear evolution equations, we find when seeking solutions for which all the derivatives tend to zero as Ixl 4~, the well known solutions : Burgers' Shock Wave (Purely dissipative)

vet)

1 + 1 + + I(V... + v...) - I(v... - v...) tanh [(v... - v...)1;/4v] •

•

(25)

satisfying lim

viti

Itl .....

•

v±

...

...

with v

>

+

v~

and a

-- - - - - - - -- ----- Uoo

\f+ ----«I:J

o KdV Solitary Wave (Purely dispersive) (26)

92 satisfying lim V(I;)

v..

11;1--

with v..

~

0

and s

z

v..

+ a/3.

o The Burgers' shock wave, as the solution (25) is called, is seen to propagate with a speed s • (v..-

+ v..+ )/2 that is uniquely determined by the

boundary conditions, but is invariant with respect to an arbitrary fixed spatial translation.

In general, all solutions v(x,t) of Burgers' equation

are invariant with respect to a Galilean transformation.

This solution links

two different constant states at plus and minus infinity. The KdV solitary wave, as the solution (26) is called, is different and is a pulse shaped wave that, relative to the same constant value v.. at plus and minus infinity, tends to zero together with all its derivatives as

11;1--.

Its speed of propagation relative to v.. is proportional to the

amplitude a, and its width is inversely proportional to the square root of the amplitude. by

In this travelling wave solution the speed is not determined

the boundary conditions, but by the amplitude a

>

O.

Like Burgers' equation,

the KdV solitary wave is also invariant with respect to a Galilean transformation. Z3busky and Kruskal found numerically that a KdV solitary like a particle.

wav~

behaves

Specifically they found that when two different amplitude

waves of this type are such that the one with the greater amplitude starts to the left of the one with theless e r amplitude, then the larger one overtakes the smaller one and , after interacting with it, the waves have merely inter-

93 changed positions.

This is a nonlinear interaction yet the pulse shapes

are preserved exactly after the interaction , though the phases of the pulses (the location of their peaks) is affected by this process.

On

account of this Zabusky and Kruska1 invented the word "soliton" for a wave that preserves its identity exactly in this sense after a nonlinear interaction.

Thus KdV solitary waves are solitons.

The recent interest in solitons derives from the fact that the KdV equation is often found to arise as a far field equation and, furthermore, arbitrary initial data for the KdV equation evolves into a train of solitons together with, possibly , an oscillatory tail.

This means that solitons are,

in a sense, fundamental solutions of the KdV equation.

An

extensive

literature now exists on this topic, and we refer to the articles and to the references contained therein, in Jeffrey and Kakutani and in the various articles by Kruskal, Lax, Ablowitz, Newell and Segur in the AMS publication Nonlinear Wave Motion listed at the end of this lecture.

Many different

types of nonlinear evolution equation have been found to possess soliton solutions and for more information on this topic we refer to the review paper by Scott, Chu _ and McLaughlin for both a good account of some of them and also for the basic references, and also to the edited collection of papers by Bul10ugh and Caudrey. 4.

Inverse Scattering The behaviour of soliton solutions to the KdV equation is suggestive

of linear behaviour and this motivated Gardner, Greene , Kruskal and Miura to try to find a linearising transformation of the type used by Hopf and Cole to transform Burgers' equation to the heat equation (see Scott, Chu, Mclaughlin).

No such transformation was found, but during their search

they discovered an important connection between the KdV equation and an eigenvalue problem for the Schr6dinger equation in terms of the inverse scattering method used in

qu~ntum

mechanics .

It is this result that has

94 come to be known as the inverse scattering method in the context of solitons. We can do no more here than outline the ideas that are involved. The basic problem to be considered is how a general solution of the KdV equation

o

(27)

subject to arbitrary initial data u(x,O) • uO(x) may be obtained.

The factor

-6 is included here for convenience, but it may easily be removed by a trivial transformation if required .

In essence, the approach to this

question by Gardner et al . proceeded as follows .

When v satisfies the

modified KdV equation v

t

- 6v

2v

x

+ vxxx •

0,

(28)

they noticed that the quantity u which is given by u

v

2

(29)

+ vx ,

satisfies the KdV equation u - 6uu + u t x xxx

•

(30)

O.

Equation (29) is a Riccati equation for v if we consider u to be given. Therefore, we can use the well known transformation which linearizes the Riccati equation, (31)

This gives

o ,

(32)

where u is a solution of the KdV equation (30).

This ia a natural extension

of the Hopf-Cole transformation since the KdV equation has a third order space derivative and it is one order higher than that of the Burgers' equation. However, if we merely use (32) in the KdV equation we obtain a complicated result that is not useful .

Now the KdV equation is Galilean invariant, and

so allows the replacemen ts u ... u - )., x ... x + 6),t, so tha t (32) can be generalised to

95 (33) This is simply the eigenvalue problem for the Schr6dinger equation for ljJ with the "potential" u, where u is the solution we are seeking . Equation (33) differs essentially from the eigenvalue problem of the Scbr6dinger equation in quantum mechanics because, as u must be a solution to the KdV equation, it is time-dependent. considered as a parameter in (33).

That is, the time should be

In other words, it is required that

(33) must hold at every instant with u(x,t) at that same instant.

Thus,

generally speaking, the eigenvalues A would be expected to be time-dependent. Rather surprisingly, after some calculation,it can be shown that they are time independent (and constant), provided u decreases sufficiently rapidly at infinity . Let us deduce the relationship between the KdV equation (30) and the Schrodinger equation (33) . O~

If we let u

gives the dispersion relation w +

becomes _k

2•

~

0 at infinity,

the KdV equation

k 3 • 0, and the phase velocity A p

For .s uf f i c i ent l y small lui, therefore, we have a plane wave

propagating in the negative direction.

For large values of lui, the

nonlinearity dominates to give rise to solitons.

The linear approximation

is also valid at infinity, since lui becomes 0 at infinity. In the case of solitons, the wave decreases exponentially at infinity and k becomes purely imaginary with k • i K ' A. KP2 > 0 • p p propagates in the positive direction.

On the other hand, in the case of

the Schr6dinger equation (33), it follows that -ljJ xx small

Iu, I

and we obtain

ljJ ~

e

±1kx

Thus a soliton

2

, A • k.

~

AljJ for sufficiently

This approximation is still

valid for an arbitrary value of u, provided k is thought to be sufficiently large. and

ljJ

For a bound state, the eigenvalue becomes A • -Kp decreases exponentially at infinity.

one bound state corresponds to one soliton. soliton solution

2


.. = 0, .. . i-I

112

jA b i j sont des fonctions donnees, c= , des coordonnees locales. ou~ 1es ai' Designons par ui(x,~), q - 0, I, ••• , des fonctions q

a valeurs

plexes, derivables, de x E X et d'un parametre numerique reel rametre reel (nomme frequence), par

~(x)

reelles ou com~

, par w un pa-

une fonction reelle derivable sur X

(nommee phase). On

definit la fonct ion

u~ • w ~ sur X par :

( 1-2)

I4 I

Nous dirons avec J. Leray

que la serie formelle

i '"

(1-3)

est une onde asymptotique

r

q .. 0

pour Ie systeme d'equations aux derivees partielles

si en reportant formellement 1-3

dans

I-I , compte tenu de 1-2

on trouve un

developpement en serie de puissance de w qui s'ecrit, formellement :

=

( 1-4)

E q .. 0

W-qFi.W'f q

dont chaque coefficient Fj est nul quels que soient x et ~ • q

Les developpements de Lax correspondent au cas particul ier :

e i ~ v qi

2 - Determination de la phase

~

(x)

•

Derivons formellement, il vient en posant au (2-1)

i 9

(x , ~)

u

i q

aiq

(x,~)

113

(2-2) Dans Ie cas des developpements de Lax les formules 2-1 et 2-2 se reduisent

a: e i~

aA V i

q

(x )

c'est-ii-dire

On trouve dans Ie cas general en reportant dans I-I

une expression de la

forme 1-4 OU les coefficients pi sont : q

(2-3)

pi

(2-4)

pi

-I

0

uoi

-

a~>'

-

">' i a~ CU I 1

1

d),- 'f a),

'r

+

a),

i) 0

+

b~1. u 0i

(2-5) Les equations 2-3 ont une solution non nulle la matrice a

i

J" ),

uoi

en x € X si et seulement si

d),f est singuliere, c'est-ii-dire si Ie symbole principal de

l'operateur L est non iniectif pour Ie vecteur covariant p = (a>. ~), ce symboIe etant suppose non identiquement nul (systeme regul ier au sens de Cauchy Kovalevski) : les equations 2-3 auront une solution-non nulle sur un ouvert

n de X si et seulement si, dans cet ouvert la phase

~

verifie l'equation aux

derivees partielles du premier ordre de degre N - dite equation eikonale du systeme : (2-6)

A(x ,

'f x)

• 0

qui s'ecrit en coordonnees locales

OU

'f x

= (V 'f)(x)

114

Une phase le~on

~

verifiant l'equation eikonale 2-6 etant supposee choisie (cf.

VI), la resolution du systeme lineaire 2-3 - puis les equations sui-

vantes - depend du rang de ce systeme. Nous etudierons Ie cas Ie plus simple ou ce rang est N-I.

3 - Cas ou

'P

correspond

a une

caracteristique simple.

Designons par A(x,p) Ie polynome caracteristique du systeme I-I , c'est-a-. dire Ie polyname des composantes

PA du vecteur covariant

p defini par :

'A

det ( a~

A(x,p)

i.

I' equation 2-6 exprime que Ie gradient

'f x de

la phase l' est une racine du poly-

nome caracteristique; supposons que cette racine soit simple, { A A (x, p) }

p -

If x

«0

c'est-a~dire

que

pour tout x E.. X

oU on a pose A

A

(x, p)

aA A(x,p)

Le determinant A est alors necessairement de rang N-l pour p - ~x • V'x ~ X i Determination de u o

'r etant choisi

~

solution de 2-6 on deduit de 2-3

()-I )

oU hi(x) est une solution du systeme d'equations lineaires homogenes

115

o

(3-2)

et V0 (x, F;) une fonetion arbitraire. Puisque A(x,tp x) es t de rang N-I, hi. est determine

a un

faeteur pres.

On deduit de 3-1 : (3-3)

(3-4)

i

et wo(x)

des fonetions arbitraires de x.

Les equations les N ineonnues

F~

a

0

sont , si

If

verifie 2- 6 ,N equations lineaires pour

u~ , a determinant nul. Les u~ ne pourront done exister, re-

guliers, que si (3-5)

o

oil on a pose (3-6)

et ou on a designe par h. une solution du systeme transpose de 3-2, e'est-aJ

dire telle que

(3-7) 3-5 s'eerit

ii. J

hi a~'>' d>. U + ~ 0 + b

j

h

ii.

On a, puisque hi (respeetivement

J

ii. a~>' J

~

U d>. hi 0

hh U + ai' >' d>. wi + b~ h. wi 0 0 ~ J 0

ii.) est proportionnel a A~ pour ehaque J

J

j

116 fixe (respectivement chaque i fixe) k

(3 -8)

A~J

(x,

i

ou Aj(x,P) designe Ie mineur de a

\D) Ix

jA PI. dans Ie determinant A(x,p) et k une i

fonction de x qui depend du choix de hi et

a avoir

k

=

h. J

(on peut les choisir de maniere

I). Or, d'apres la loi de derivation d!un determinant

dA(x,p) dPA

(3 -9)

= A~J

a

(x,p)

dPA

(a~lJ p ) lJ

1

D'ou (3-10)

•

x (x, f

k A

x )

On deduit de 3-10 que l'equation 3-5 s'ecrit

•

(3-1 I)

AI. __d_ dX"

0

est la derivation Ie long du rayon correspondant

\f(x) • cte • On a pose

60 est nul si w~ • 0 (3-13)

ne depend que des equations donnees e t du choix de Calcul de

'f .

B

Une expression interessante de Best obtenue aisement Lenme

on a, si k • I B(x) = .!. 2

(3-14)

+

'1.

a ax

1.(

A A x,

1 -2

Ih • J

x

)

., "hi

aJi\

i

_0_

di

+ _

h

I dai

i -

h (---j ~

dx"

h i a.j 1

oh' I :.::r ox

+

a

la surface d'onde

117

Preuve

un simple calcul, puisque alors

Corollaire

si l'operateur differentiel L : L

est autoadjoint, on peut choisir hi de sorte que

B se

reduise A

(3-15)

= 0,

L'equation de propagation 3-1!, avec 00 2

de conservation pour U o

se

~eduit

alors A une equation

Ie long des rayons.

Preuve: Lest autoadjoint si, pour u et v A support compact on a l'egalite . . sca I a1res L2 d es prod U1ts

(v,Lu)

(u,Lv)

'A

c'est-A-dire si les matrices a A I a '2;:; On

a alors h et

jours k

=

(a.1 J ) sont symetriques et si

a.i A _ b.i 1 .

1

=

0

h proportionnels. On peut choisir h tel que h = h (avec tou-

1).

L'equation 3-11 peut alors s'ecrire

o

(3-16)

Remarque

D'apres les regles classiques de derivation d'un determinant on a:

(3-17)

si x

'1':

= x(t,y)

verifie Ie systeme differentiel des rayons associes

a

la phase

118

et on a designe par

d

de·

~

()

A ()x~

la derivation le long de ces rayons.

L'equation 3-16 exprime la conservation le long des rayons de la densite de

dt d

[u2o D(y)] D(x)]

0

119

III ONDES ASYMPTOTIQUES D' ORDRE q, ONDES APPROCHEES. EXEMPLE.

I - Determination des termes successifs u i

q

Uo(X'~) etant determine verifiant 3-5 de la le~on lIon pourra trouver u~ verifiant pj • O. sa forme generale sera o (1-1 )

ou

VI(x.~)

est une fonction arbitraire et

i

VI(x.~)

est une solution des equa-

i et U

des primitives par

tions lineaires (1-2)

On deduit de I-I en designant par rapport

a~

u(x.~)

(x.~)

de V et Vi :

(1-3)

Supposons alors construites des fonctions u i • p < q • verifiant les equations pjp (1- 4)

=

0 • p < q-I et soit uqi de la forme u

i q

p

120

ou U

q

i (x.~) es t arbitraire et U est une solution quelconque, mais fixee . des q

equations lineaires

tiqi

(1-5)

+

gi

q-I

'). i _ a1.,J ~,u

• 0

1\

q-I

. i + b.J u

1.

q-I

i i Pour ces fonctions up , u on a aussi q

o

(1-6)

Les equations Fi • 0 s'ecrivent q (1-7)

.i uq+ 1 +

a . iA d). 1.

gi - a J' A d u i i A q q

gi • 0 q

+

b~1. uqi

on pourra t rouver u. i + 1 S1.· e t seu I ement S1.' q

o

(1-8)

Cette equation est une equation differentielle du premier ordre lineaire pour Uq ' analogue

a

II. 3-5. qui s'ecri t :

k AA d U + B U + cS q q A q

(1 -9)

ou Best la fonction de x

est connu quando

0

II - 3-13 et :

u~_1 e tant determine . U~ a ete choisi .

2 - Ondes approchees .

Nous dirons que Ie developpement fini

121

(2-1)

u

i

r

E p -

o W 0

est onde approchee d'ordre r -I si il existe une constante M telle que pour tout w

(2-2)

ou

j

IlL (u)1

Ix designe

{2-3) On

une norme convenable, par exemple

IILj(u)ll x

Sup xES: X

{I)

ILj (u) I

voit que la condition 2-2 sera realisee si les u

j i verifient F • 0, P < r p p

et sont bomes, ainsi que leurs derivees partielles par rapport aux variables

x. D'apres les equations )-7 les

u~+)

dependent lineairement des

ra trouver une onde approchee d'ordre r-\ si u

a t;

i o

u~

; on pour-

admet r primitives par rapport

uniformement bornees , pour t; Eo. R, ainsi que leurs derivees partielles par

rapport aux variables x . Cette condition pourra toujours etre verifiee, par un choix convenable d'une donnee initiale, au moins dans un domaine de regularite des rayons . Cette construction d'ondes approchees d'ordre quelconque ne sera en general pas poss ible pour les equations non lineaires.

3 - Exemple

ondes dans les flu ides compress ibles )-dimensionnels linearises.

Les equations classiques non lineaires sont

(I) Pour les equations hyperboliques on est amene normes .

a

considerer d'autres

122

ou t est Ie temps, x la variable spatiale. p , n et u respectivement la dens ite, la pression et la vitesse. Les equations lineairsees au voisinage d'un etat de repos sont

(3-1)

ou r et a sont des fonctions donnees de x. Cherchons une solution asymptotique approchee d'ordre I, avec

(3-2) v • v 0 (x, e,

W

'f

'f(x,t) :

I 'f) + w v I (x, t.w If )

on trouve pour l'annulation des termes en w,avec 'fx· 3tr / 3x, 1ft • 3'f/3t F

I -I

F

2 :: r V• ~ t + Po -I o

-

(3-3)

On

pourra verifier

Po \ft + a

I

rvo

\.f x

'fx

0 0

F~I • F: 1 • 0 avec Po et Vo pour tous deux nuls si l'e-

quation eikonale suivante est verifiee :

(3-4) II Y a ici deux familIes de phases possibles qui sont si a est une constante \F(x,t)

= ~(x

- at)

et

f(x, c)

~(x +

at)

Les rayons correspondants etant alors les droites x - at • c~ et x + at • c~~ si a est une fonction donnee de x les deux familIes d'ondes sont obtenues par

123

resolution des equations lineaires du premier ordre

'ft - a 'fx • 0

(3-4 b)

et

(3-4 a)

Les rayons spatio-temporels correspondant

a

la premiere fsmille sont les cour-

bes solutions de l'equation differentielle dx dt

a(x)

soit x • f(t,y)

f(o,y) • Y

Ie rayon issu d'un point (o,y). Supposons l'equation inversible (c'est toujours Ie cas pour t assez petit puisque f(o,y) • y) on en deduit que si (x,t) est un point du rayon issu du point y on a : y

•

g(t.x)

g(o.y)

y

La solution de (3-4 a) telle que (3-5)

If(o,x)

•

t/J(g)

au t/J est une fonction donnee, est (3-6 a)

puisque

'r(t,x)

If est

•

t/J(g(t.x»

constant sur Le r!1yon spatio-temporel (bicaracteristique de

l'eikonale) issu de y. De meme la solution de (3-4 b) telles que (3-5) soit verifiee est

ou g-(t,x) est obtenu par inversion de la solution de dx dt

•

- a (x, e)

La solution generale des equa tions pour un

tp

de la premiere fsmille est

124

(3-7) (on a integre en

et supprime Ie terme independant de

~

O W est equivalente

L'annulation des termes en

~

).

a:

(3-8)

Pour

If2t

- a

2

r 2x ces equations impliquent

(3-9)

rempla~ant V

o et Po par 3-7 on trouve l'equation de propagation pour Uo' de

la premiere fami lle

(If t

= -

a

r x)

(3-10)

equation de conservation Le long du rayon du facteur a r de

rl-o ,

independanment

~ .

On

resoud ensuite les equations

ou PI' VI

F~ - 0, i = 1,2, sous la forme

est une solution particuliere, par exemple, U satisfaisant o

a

1-10

d'ou ou U designe une primitive de 0 Si on calcule Lu

=

Uo(t,x,~)

par rapport

a

~

.

2 (L (p,v), L (p,v)) pour la somme 3-2 ou Po' PI' v o' vI I

sont les fonctions qu'on vient de determiner on trouve :

125

I L (p,v)

1. F 1

-

w

2

1

=

1. (a w t PI

1. F 2 =1. (r at w 1 w

L (p,v) -

+ a

2

r ax vI)

+ ax vI)

donc ILi(P,V)I

~

-1

w

si les derivees partielles en t et x de ceci sera realise si et

~,

Uo(t,x,~)

et ceci pour tout

Uo satisfaisant

a

~

et

M PI(x,~,~)

Uo(t,x,~)

vl(x,t,~)

sont bornees :

sont uniformement bornes en t, x

E R.

l'equation de transport 1-10 (condition de conservation)

sera uniformement borne en

~,

dans tout domaine du plan (x,t) ou les rayons

ne representent pas de caustiques (equations x inversibles) s'il en est ainsi est :

et

a

= f(t,y)

donnant les rayons

l'instant initial taO. Un choix possible

126

IV DEVELOPPEMENTS ASYMPTOTIQUES POUR DES

E~ATIONS

QUASI LINEAIRES A CARACTERISTIQUES SIMPLES.

r Definitions

m

Nous designons par X une variete differentiable de classe C de dimension n ~.

• • de par x un p01nt

X.'

xO.x l

•

~ l I es d e x. •••• x ~-! sont d es coord onnees oca

Nous considerons un systeme de N equations aux derivees partielles du premier ordre sur X. quasi lineaires. aux N inconnues uk(x). fonctions(!) sur X

a

valeurs complexes (1-1 )

en posant u

L u

=

I

u ,

••• , u

N

o

et en des i.gnanz par u ... L u une application (non

lineaire) de l'espace des suites {uk(x)} de N fonctions differentiables contenues dans un polycylindre P :

(I) les inconnues uk(x) peuvent aussi etre des tenseurs sur X. l'application

u » Lu s'ecrit encore !-'3 en coordonnees locales. les u i etant les composantes des tenseurs envisages,

127

uk(x) fonctions donnees ,

(1-2)

o

dans l'espace des suites de N fonctions, ayant dans chaque systeme de coordonnees locales une expression de la forme 'A i _ a~ (x,u) au, + bj(x,u) 1. ax"

(1-3)

tx

s

0

i,j

s

I, •• ,N; A = 0 ••• ,t-I

.

at (x.u) et bJ(x.u) sont des fonctions de classe C~ en x, analytiques en u sur P et egales

a

la somme de leur developpement en serie entiere

(1-4)

bj(x,u) ou on a pose

Designons par u

i q

• q - 0.1 •••• , des fonctions

(x.~)

vables. de x €X et d'un parametre numerique reel (norome frequence), par

~x)

~

a valeurs

complexes. deri-

• par w un parametre reel

une fonction derivable sur X (noromee phase).

Posons (1-5)

u

i

u

q

i q

0

w

i u (x, q

w'f' )

Nous dirons comme dans Ie cas lineaire (cf. II) que la serie formelle (1"-6)

u

i

est une onde asymptotique pour le systeme d'equations aux derivees partielles si en reportant formellement 1-6 dans 1-3, compte-tenu de )-4 on trouve un developpement en serie de puissance de w qui s'ecrit, formellement :

12 8

( 1-7)

w-q

E q

=0

dont chaque coefficient F

j q

F~

• W

If

=

0

est nul que1s que soient x et

2 - Determination de 1a phase

~

~.

•

On pose. comme dans I I .i

( 2-1)

u

q

on a (2-2)

On trouve a10rs en r eportant dans \-3 une expression de 1a forme \-7 au les coefficients F

j

q

sont

(2 -3)

(2-4)

j F 0 F

(2-5)

j I

(ui a 'f - a~A 10 I A

-

a

+ Cl " u i ) +a jA uh u.i Cl 'f + b j o 1 0 A 0

j A •i j" h i. i h .i jA i + j i u i o 2 Cl A'f + a i o Cl" u 1 b i u \ +aih{u l(Cl "Uo+ul Cl,,' f) + u2 uO ClA'f}

129

Nous etudierons Ie cas ou

U

i

o

est une solution donne e, i ndependant e de

S,

des

equations 1-3. (2-6)

Les equations Fj • 0 s'ecrivent alors q

(2-7)

j F-I

(2 -8)

F

(2-9)

F

j 0

j 1

-

0

.i u - a~A 10 CIA 'f 1

-

a~A

ClA~

10

• 0

.i + a jA CIA u i + a jA CIAIf uh u.i ih 1 1 2 1 io

U

'A

i

j)

+ (aih CIA u + b 0 h

(2-10)

F

j q

-

u

h 1

.

0

a~A CIA If' u.1'+ + a jA CIA ui + a jA CIA ih q io 10 q 1 'A

+ {alh (CIA

OU fj ne depend que des u i q

p

,

.i u p

'P

'f

h .i u u 1 q

. i + CI ui) + b j } uh + f j • 0 u1 A 0 h q q

, CIA upi OU p < q.

.i L'equation 2-8 entraine, si l'on veut que u t 0 1

(2-11)

A(x,

'f' x)

o

2-11 est une equa tion aux der i ve es part ielles du premier ordre qui exprime

que la phase

'f

est solut ion de l' equation caracteristi que approchee, obtenue

en donnant aux coefficien ts principaux de 1-3 la valeur qu 'i1s prennent pour la solution donnee ui(x) . o On designe par A(x,p) Ie polynome caracteristique du systeme 1-3 correspondant

a

la solution

u~ , c' est a dire Ie polynome des composantes PA du vecteur

covariant p defini par : A(x,p)

' A p,)

det(a~

10

J\

130

I' equation 2-11 exprime que Le gradient

'f x

de la phase

'f

es t une racine de

ce polynome et on suppose que cette racine est simple, c'est pour tout

a dire

que

x Eo X

ou on a pose

aA A(x,p) Le determinant A est alors necessairement de rang N-I pour p =

l'

'f x'

Vx Eo X.

etant choisi solution de 2-11 on deduit de 2-8

(3-1) ou hi(x) est une solution du systeme d'equations lineaires homogenes

(3-2) et

o

VI(x,~) une fonction arbitraire . Puisque A(x, fx) est de rang N-I. hi est

determine

a un

facteur pres.

On choisit : (3-3)

(3-4) Les equations F

j 1

.i les N inconnue s u 2

=0

sont. si

If

• a determinant

verifie 2- 11, N equations lineaires pour nul. Les u.i ne pourront d onc exister 2

,

131

reguliers, que si (3-5) ou on a pose (3-6)

et OU on a designe par h. une solution du systeme transpose de 3-2, c'est J

a

dire telle que h. J

a~).

a, ID T

designe Ie mineur de

af~

p). dans Ie determinant A(x,p), et k une

(3-7)

1.0

•

0

1\

On a (cf . II)

(3-8) ou

A~(X,P)

fonction de x qui depend du choix de hi et

a

h.J

(on peut les choisir de maniere

avoir k • I) et

(3-9)

On deduit de (3-9) que l'equation 3-5 s'ecrit A aUI (x,!:;) k A A

(3-10)

ax

+ a(x) UI(x,;) UI(x,;) + B(x) UI(x,;)

est la derivation Ie long du rayon correspondant

a

E

0

la surface d'onde

cte

On a pose (3-11) (3-12)

i

a et Bne dependent que des equations donnees et de la solution u o•

132

Le coefficient

ea

une expression analogue

aire, en 11.3.14, une fois la solution u R. 11 .3.14 par b i + aR.i a). Uo j

Le terme aU

j).

I

01

i o

a

ce11e obtenue dans Ie cas line-

rempla~ant b~1 dans

choisie, et en

n'existe pas dans Ie cas lineaire. La non nullite du coef-

ficient a va provoquer un effet, lie

a

la non linearite, de dis torsion des

signaux. Pour ealculer Ie coefficient a on remarque que a

(3-13)

•

or, d'apres 3-8 on a (3-14)

=u • 'f~

en eonsiderant que

on demontre (cf Boillat ~ • d er ive e

On

')'

A(x,u,p)

(3-15)



(t(x), y(x»

hi(x)

c'est-a-dire que ce terme es t, comme dans Ie cas lineaire, proportionnel au vecteur propre a droite hi de la matrice A(x, facteur de proportionnalite WI d'une fonction

cI>

cI>

~x) (valeur propre zero). Le

est, comme dans Le cas Li.nea i t e , produit

qui ne depend que de l'equation et de la surface d'onde don-

nee (et ici de l a solution u

i)

o

et que l'on calcule par integration Ie long

des rayons correspondants par une fonction WI' dite facteur de forme, qui depend d'un choix initial. Cette fonction est, dans Ie cas lineaire, constante Ie long des rayons; elle ne l'est plus dans Ie cas non lineaire, si a n'est

136

pas nul : nous dirons que la non linearite provoque en general une dis torsion des signaux, et nous enoncerons Ie theoreme : I. Le long de varietes caracteristiques exceptionnelles les signaux se pro-

pagent sans dis torsion. 2. Si Ie facteur de dis torsion a est de signe constant il existe un temps critique au dela duquel la distorsion du signal conduit

a

la disparition de

l'onde asymptotique (formation d'un choc).

References.

[11

Leray J., Particules et singularites des ondes ..• Cahiers de Physique, t.

[2J

IS, 1961,

p

373-381.

Garding L., Kotake T."

Leray J., Uniformisation et solution du probleme

de Cauchy lineaire ••• Bull . Soc. Math. 92, 1964, p 263-361.

[31

Lax P., Asymptotic solutions of oscillatory initial value problem, Duke mat . J., 24, 1957, p. 627-646 .

[4]

Ludwig D., Exact and asymptotic solutions of the Cauchy problem, Comm. on pure and appl. Math. 13, 1960, p 473-508.

[5J

Choquet-Bruhat Y., Ondes asymptotiques pour un systeme d'equations aux derivees partielles non lineaires, J. Maths pures et appliquees, 48, 1969, p. 117-158.

[6]

Anile A.M. and Greco A., Asymptotic waves and critical time in general relativistic magnetohydrodynamics, Ann. I.H.P . vol XXIX,nD3, p. 257,197&

[7J

Boillat G., Ondes asymptotiques non lineaires, Ann. Mat. pura et Appl.

iL,

1976, p. 31-44.

137

V APPLICATION AUX EQUATIONS DES FLUIDES PARFAITS RELATIVISTES.

I - Equations des fluides parfaits relativistes .

En l'absence de courant de chaleur, les equations s'ecrivent (1-\ )

(1-2)

(1-3)

(p + p) a

u

U

a

'V

a

uS _ yas

a

a

a

d p+ (p + p) 'Va u a

ua d S a

p

K

K

.

0

0

(E)

0

p , p et S sont respectivement la densite d'energie , la pression et l'entro-

pie spec i f i que , Ce sont des "grandeurs d'etat" fonctions donnees de deux d'entre elles, par exemp1e p et S pour 1'equation d'etat ( 1-4)

On

p

p(p , S)

sait que si l'on pose p

r(I + e)

ou rest 1a densite materie1le propre et e 1'energie interne specifique,

138

1-3 est, compte-tenu de l'equation thermodynamique de • T d 5 + P r- 2 dr equivalente

a

l'equation de conservation de la matiere introduite par Taub [2]

On a pose

a,S·0 , 1,2,3

(1-5)

ou gaS est une metrique hyperbolique donnee ; V designe la derivee covariante ex correspondante et d ex On peut ecrire les equations 1-2, 3, 4, sous la forme

I,J· 0,1, • •• , 5

(1-6)

en posant u u

I

4

- u

a

pour I • 0, •• • 3

- P

2 - Determination de la phase

If .

Cherchons une solution asymptotique des ces equations, au voisinage d'une solution donnee uo' Po' So' c'est

a dire

perturbe d'un mouvement donne. Soit

un mouvement vibratoire du fluide,

139

P

S +

S

o

L q =

en supposant que l'equation d'etat p • p(p , S)

est une fonct ion analytique de P et S, et on ecrit (2-1)

p(p,S)

Po + P~

(p - po) + Ps

o

(S - So) + ••• 0

en posant

Les termes independants de w donnent

, S

0, • • • ,3

(2-2)

On a pose

Ces equations lineaires en ~~

PI'

81 n 'ont

une solution non nulle que si

Ie determinant A est nul. On trouve pour A, comme prevu, Ie determinant caracteristique du systeme (E)

ou on a pose

140

a

~

a 'P o Ia

u

La condition A ~ 0 exprime que les varietes ~x) • c t e sont des varietes

a dire

caracteristiques, c'est solution

des surfaces d'onde, des equations (E) pour la

u~, ondes acoustiques (a 2 + b p' Po

0) ou ondes materielles (a • 0).

3 - cas ou la phase verifie l'equation des ondes acoustiques .

Supposons que (3-1 )

A

A est, pour A

-

bp' + a Po

aB :: {p' g - u uo(p~ 0 po· 0

2

-I}'fa 'fB

=

0

a u l ' PI' 51) est donne par

I

0, de rang N-I • 5, {u! }

(3-2) ou U1(s,x) est une fonction scalaire et les hl(x) sont une solution de 2-2, c'est a dire proportionnels aux mineurs d'une meme ligne de A. Calculons, par 1

exemple les mineurs de la quatrieme ligne A on trouve : 4,

A4 • (p + p)4

4

0

as

Nous prendrons hA _

aA

I

pp Yo 0

(3 -3)

'fa

h4 • (p + p) 0 a 5 0 h

141

Appliquons la methode generale pour trouver l'equation ditferentiel le verifiee par U en egalant

:

I

a

zero les termes en w-I des equations on trouve

(3-4) o~

(l'indice I indique que l'on prend dans la quantite ecrite Ie coefficient -I

de w )

a s[( Pp') I P• ; + (Ps ') I - Yo

51]

'fa + (p + p)o uao

M v

a

uSI

(3-5)

Les equations 3-4 ont une solution

·1

z si

U

(3-6)

ou les pIe

hJ sont proportionnels aux mineurs d'une mieme colonne de A, par exem-

a~ .

On

a

A4

A

4 A 5 On

prendra

4

(p + p)o a

as - Yo

4

'f a PS'

0

4 A 4

'fA 4

(p + p) 0 'fs a

3

=

4

( p + p)o a

5

142

(3-7)

a

J

Formons hJ gl en

2

u

I

I

trouve que U doit satisfaire 1 l'equation differentielle du premier ordre suivante rempla~ant

1

par U h • 1

On

(3-8) avec

(ACl est

=-

k

(3-9)

;.a

2

2 a (p + p)o

1 aX • '2 dpex

bien Ie rayon acoustique)

et, compte-tenu de 3-1 : 2

ex = ~ {2(1 - p' ) 2 Po

(3-10)

(p + p)o )

alors que -I

u

+ pIt

ex

o

(p+p) 0 -

2

(P+P) _0

2 p' P Po -I (p+p)o -~{PS Po

oil on a designe par

a

I' operateur des ondes acoustiques

ex" + uex }) V :: (p' Yo 0 0 ex Po 23 A r:; rap aPA} ex a" ) ex PA = 'fA

O'f (on a

a

J

=

'2

Si la solution de base es t

a densite

~"

et entropie constantes, et

a

lignes de

143 courant

a divergence

nulle

o

(3-12)

a se

reduit

o

o

a

a •

t

Q

!p a

on remarque que les conditions 3-12 impliquent, si u o' Po' Po' So est une solution de (E) que ua Va u a • 0 o

0

'

d'ou on deduit que

Remarque

a est nul si • 0

et c'est

a dire

viste cf

en particulier si Ie fluide est incompressible (au sens relati-

[IJ),

c'est

a dire

admet l ' e qua ti on d'etat p • p + cte

Dans ce cas les andes acoustiques sont exceptionnelles au sens defini preced eement ,

Pour les fluides reels on a p' ~ P

1

p" p

>, 0

donc a

~

0

Si p'p < 1, ou p"p > 0 on a raidissement des signaux. On peut aussi trouver un

dev~loppement

asymptot ique correspondant aux ondes

materielles; celles ci apparaissent comme multi ples (en accord avec la theor ie genera Ie de Boillat, cf [4J). Le probleme de Cauchy oscillatoire est

144

reso1ub1e, au premier ordre d'approximation, pour 1es equations des fluides parfaits. II n'en est plus de meme en presence de phenomenes dissipatifs, par exemple pour un fluide charge

a

conductivite non nulle (cf [3

J) .

References.

[.]

Lichnerowicz A., Hydrodynamique et magnetohydrodynamique, Benjamin 1967.

[2]

Taub A.H., High frequency gravitational waves and average lagrangian, General Relativity and Gravitation, Einstein Centenary volume, A. Held ed, Plenum.

[3}

Choquet-Bruhat Y., Ondes asymptotiques pour un systeme d'equations aux' derivees partielles non lineaires, J. Maths pures et appliquees, 48, 1969, p . 117-158. Coupling of high frequency gravitational and electromagnetic waves, Actes du Congres Marcel Grossmann, Trieste, Juillet 1975).

r4}

Boillat G., Ondes asymptotiques non lineaires, Ann. Mat. Pura et Appl. ~,

[51

1976, 31-44.

Anile A.M. and Greco A., Asymptotic waves and critical time in general relativistic magnetohydrodynamics, Ann. I.H .P., vol.XXIX, n03, 1978,p .2S7.

[6]

Breuer R.A. and Ehlers J., Propagation of high-frequency electromagnetic waves through a magnetized plasma in curved space-time, to appear in Proc. Roy. Soc. A.

145

VI DETERMINATION DE LA PHASE. BICARACTERISTIQUES. VARIETES LAGRANGIENNES.

I . Definition

des varietes lagrangiennes.

L'equation eikonale 1i laquelle doit satisfaire la phase (I-I)

A(x,

'fx>

If:

= 0

est une equation aux derivees partielles du premier ordre non lineaire, A est un polynome de degre N, homogene, en

~x'

Nous allons rappeler comment on in-

tegre une telle equation, en utilisant Ie langage de la geometrie symplectique. On designe par T*X l'espace fibre cotangent

a

la variete X de dimension ~.

Un point de T*x est un couple (x,p) ou pest une I-forme sur l'espace tangent TxX

a x en

x, c'est

a dire

un vecteur covariant.

Une solution de l'equation )-1 dans au-dessus de

n C.X est une section du fibre T:tX

n: par

telle que

(d

'f) x

• < p, dx >

x .... (x, p , Ip(x» A • 0, • ••

~-I

x lR,

146

et

o

A(x, p)

sur une te11e section on a

Soit IT : T~ + X la projection canonique (x,p)'+ x de T~ sur X. On definit une (-forme sur T~, appelee I-forme fondamentale ou forme de soudure par: B(

x,p ) (u)

c

u €.

p(lI' (u»

T T*x X,p

son expression en coordonnees locales est : B

Ph dx

x

La 2-forme

(j

est fermee et de rang (j

2~,

=

de

une telle 2-forme est dite symplectique. La 2-forme

munit T~ de sa structure symplectique canonique . Une sous-variete de T*x

qui annule

(j

et qui est de 1a dimension maximum possible, c'est

a dire

:

~

,

est dite lagrangienne. Si Vest une variete de dimension ~, immergee dans T*x par une application f, on dit que (V,f) est une sous-variete 1agrangienne [immergee] de T*x si

o

sur V

ou

f

2 Recherche d'une sous-variete lagrangienne (V,f) de T~ telle que A(x,p) = O.

Le prob1eme est celui de la recherche des varietes integra1es (immergees

147 dans T~)de dimension ~ du systeme differentiel exterieur

o

a

(2-1)

La fermeture de ce

s~steme

o

(2-2)

contient, outre 1es equations precedentes, 1'equa-

tion exterieure sur T~ :

(2-3)

o

dA

Le systeme caracteristique de 2-1, 2-2 est Ie systeme associe de 2-1, 2-3. 11 est constitue par les champs de vecteurs v sur T1x tels que : (2-4)

i

v

a

k d A

avec

k€:JR

c'est a dire en coordonnees canoniques (XA,PA) de T*X ou v A - 0, ••• ,

= (vA,VA+~)

~-I

(2-5) Un champ de vecteurs vA possedant la propriete 2-4 est dit hami1tonien pour 1a structure symp1ectique a et 1'hamiltonien A. On remarque que vA est tangent

a

Jl: (sous-varie te

A(x,p)

0 de T1x)

Une trajectoire du champ de vecteurs hamiltonien vA est appelee une (courbe) bicaracteristique de l'equation aux derivees partielles A(x,

~x) =

o.

On suppose que l'hamiltonien A n'a pas de point critique sur ciC(c'est

dire que dA sur

en: , et

~

0 quand A

=

a

0), Ie champ hamiltonien vA n'a alors pas de zero

on d.emontre Le theoreme fondamental suivant (cf par exemple

Ill):

Soit Y une sous-variete compacte de dimension t-I de T*x verifiant a

=0

et A

= 0,

transversale en chaque point au champ hamiltonien vA. Soit f t

Ie flot du champ de vecteurs vA' alors (Y x :JR ,f)

ou

f: Y x :JR ... T~

par (y,t) ..... ft(y)

14 8

est une sous-variete 1agrangienne immergee de T~.

3. Determination de 1a phase.

Etant donnee une sous-variete 1agrangienne (V,f) immergee dans X, il existe dans chaque ouvert U C V simplement connexe une phase constante additive pres, satisfaisant

a

~

, determinee

a

une

l'equation

(3-1 )

puisque l'on a, sur U, fX de = fXo

= O. v

Remarque : 11 existe toujours, globalement, sur Ie recouvrement universel V de V, projete sur V par

v

IT, une fonction

v ~

v d ~

On deduit de IT

0

~

(f 0

telle que v IT) x

, connu dans uc V, une phase

~

e dans n ex si l'application

f : U ~ nest inversible.

L'application f etant une immersion il existe toujours un sous-ouvert, encore note U, de U, tel que f soit un diffeomorphisme de U sur feU). L'application IT

0

fest alors inversible sur U si la projection IT : T*X ~ X restreinte

a feU) est inversible : il en sera ainsi au voisinage de tout point ou feU) n' est pas tangent

a

la fibre de TXX, c'est

a dire

n'a pas un plan tangent

"vertical". Soit x€.n

ex,

tel que IT-I (x) ne soit pas tangent

1es points de feU) tels que IT(Yi) = x

a

feU). Soit Y1, .. • Yk

149 Le poin t x admettra un voisinage dans X, encore note n tel que IT-Icn) soit l'union disjointe d'ouverts de fCU) :

C3-2)

U

i

A chaque U, pour une meme phase

~

= I, •.• ,k

fCUi)

sur V, correspond une phase

~

i sur

n

donnee

par :

If i

= ~ 0 (IT

i

0

f)

-)

,

Dans les applications la donnee physique est souvent la variete lagrangienne V, provenant de la geometrie du probleme et de sa dynamique : les bicaracteristiques, c'est

a dire

les trajectoires du vecteur hamiltonien vA sont les

rayons lumineux Cdans l'espace des phases) dans les problemes d'optique, les trajectoires des particules materielles dans d'autres problemes. Nous allons cons i d e r er Ie cas ou la projection de V, supposee sous-variete de simplifier, sur Q

c: r%x

-I IT cm = V ........ L =

=

U

i = 1, •• • k

oU chaque res tric tion

pour

n'est pas bijective, mais OU il existe un sous-ensem-

ble L de V (son "contour apparent") tel que IT-I cm (3-3)

r*x

1\ de

IT

V " L soit de la forme

U.

L

a Ui IT .

L

U.

L

~

Q

est un diffeomorphisme.

4. Solutions asymptotiques.

II est nature 1 de chercher une solution asymptotique du systeme differentiel

150

d'equation eikonale A(x,

~x)

= 0,

correspondant a une variete lagrangienne du

type 3-3 sous la forme : k

(4-1)

E L •

ufx)

ou les i'i sont des phases, sur

i,

e

n c. X,

w 'f i (x)

vi (x)

correspondan tala varie te

v.

On

sera

aide dans ce calcul par la methode de la phase stationnaire qui montre (cfVII) comment 4-) liee a l'evaluation asymptotique d'une integrale. Des developpements du type 4-1, et les equations de transport correspondantes, sont utilisees pour determiner l'intensite lumineuse en presence de caustiques (enveloppe des rayons lumineux en projection sur l'espace-temps X). Remarque : Chaque phase

~i

n'est connue qu'a une constante additive pres,

puisque la variete lagrangienne V ne determine

Wqu'a

l'addition pres d'une

constante, depourvue de signification physique. La theorie des integrales asymptotiques et de l'incide de Maslov permet de determiner des relations entre ces constantes, puis des conditions sur la variete V, dites

cor~itions

de quan-

tification de Maslov, pour qu'il corresponde a V une solution asymptotique globale, avec une phase

[3l

et

[51

de VIII).

\f

determinee globalement sur n

= n(V)

(cf references

151

VII PHASE STATlONNAlRE. PARAMETRlSATION D'UNE VARlETE LAGRANGlENNE.

I. Methode de la phase statiounaire (une variable).

On se propose d'evaluer, pour w grand, une integrale de la forme

lew) -

a

00

ou a et f sont des fonctions C , et a est J

O

)

On suppose que ~~

support compact.

ne s'annule pas sur Ie support de a; on deduit alors

de

a aa

(e i w f)

que

lew)

I

z

iw

•

iw

l'integrale etant bornee par un nombre M independant de w on a -I

Mw

152

Par iteration du procede on trouve, pour tout n

~ ~

0 (w- n)

I(w)

2°) On suppose que f s'annule en un point et un seul que

a2 f ---2 r ao

0 pour

0

= 0

0,

a dire

c'est

0

0

du support de a et

que f a un point critique, et un seul.

non degenere sur Ie support de a . On montre que f peut alors s'ecrire dans un voisinage de

ment de variable

o~

=

g(o) +

I(w)·

e

iwg(o)

t t2

J::

donc

=

0,

par un change-

t(o) tel que t(oo) - 0, sous la forme

g(t) - f(o(t»

_ ou b I t )

0

e: •

e i we:/2 t

2

b(t) dt

do a(o(t» dt

l'integrale existe, absolument convergente, puisque b a un support compact. On peut en particulier la calculer par passage

I(w)

A

f-A

lim A = +

00

eiwe:/2 t

2

a

la limite

b(t) dt

On pose alors b I t ) = b(o) + t c(t)

On sait que lim A=oo

+A

f-A

b(o) e

iwe:/2 t 2

dt

b (0)

On va. estimer I I (w) ::

lim A=+oo

+A

f

eiwe:/2 t

2

c (c)

t dt

-A

Puisque b a un support compact on a c(A)

c(-A) done

(2II») /2 e iIIe:/4 w

153

I(

(

W

)

_1_

m

iwe:

+A

lim

A =

J-A

+ (x.ai(x»

de l'equation de Cf , ~~ (x, a) localement) si

=

est determine par resolution en a

O. Cette resolution est possible (au moins

d2 f

---2 (x ,u) f O. L'ens emble singulier L de la variete lagrandU

gienne V (cycle de Maslov) est l'image par ~ des points de X x :JR •ou af = 0 da 2 d f

et - -

dcl

= O.

La definition d'une parametrisation d'une variet~ lagrangienne s'etend en rempla~ant

X' x :JR par un espace fibre de dimension £ + m au dessus de X (par

159

exemple T*x dans lequel Vest deja plonge). L'obtention d'une parametrisa tion donne la pos s Lbi l i.t.e de fixer dans l' ouvert n C dant a chaque U diffeomorphe par i

ITlu.

a

n

X une phase

'f'i cor r esporr-

et de remplacer 1a valeur asymp-

1

totique d'une integrale du type 3-1 en uti1isant 3-2 par une somme de termes faisant intervenir ces phases. Exemple : supposons que nous ayons une seule variable d'espace x et une variete lagrangienne dans Ie plan (x,p) qui passe par Ie point (0,0), et est tangente en ce point a l'axe des p. p

v

11 est alors naturel pour "deplier la singula-

rite", c'est a dire i c i pour trouver la var Lete et la fonction f(x, a), de chercher cet en-

C

f

x

semble C dans X x lR sous la forme : f

x ce sera 1 'ensemble

Ct

2

~. 0 si f est de la forme aCt f(x,Ct)

a(x) +

XCt -

3a

3

f sera une parametrisation de V si af ax

a'ex) + a

~

est tel que l'on a

V-

{x, a'ex) + Ct} x

= a2

c'est a dire si Vest compose des deux ouverts u\ =

{x, a'(x)

Ix }

x > 0

U2 =

{x, a'(x) - Ix }

x > 0

+

et

et de l'ensemble singulier : L

(0,0)

160

VIII DEVELOPPEMENT ASYMPTOTIQUE AU VOISINAGE D'UNE CAUSTIQUE .

I. Caustiques du premier type.

La forme d'une parametrisation d'une variete lagrangienne est liee ture des singularites de l'inverse de la projection

n:

a

la na-

T*x + X restreinte

a

V. Les singularites des applications differentiables ont ete classifiees par Thom. On montre que dans Ie cas Ie plus simple ( pliage, auquel correspond l'exemple

a

la fin de VII) on peut parametriser V par une fonction de la

forme : f(x,a)

o(x) + p(x)a -

3

a :r

ou a et p sont des fonctions regulieres du point x E X. L'ensemble C est alors f

~~ _ p(x) - a 2

0

. a2 f L'ensemble singulier E de Vest l'image de l'ensemble ---2 II se projette sur X en p(x)

=0

u e; ±

et on a

aa

=- 2a

=0

de Cf'

161

La, figure represente la projection sur

X : l'ensemble singulier E se projette sur l' ensemble C appe Ie "caustique". L'ensemble p(x) < 0 n'est la projection

p(x) < 0

d'aucun point de V. l'ensemble p(x) > 0 est recouvert deux fois par la projection de V : deux "rayons" passent par chacun de ses points.

2. Integrale asymptotique. fonction d'Airy.

Pour construire des solutions asymptotiques d'un systeme differentiel correspondant

a une

variete lagrangienne du type precedent on va etudier la va-

leur asymptotique de l'integrale u(x) •

f e iwf(x.a) a ( x.a ) da = - e iwo(x)

fe

iw(p(x)a

3

~

ex 31)

a(x.a) da

On sait que (theoreme de preparation de Malgrange) il existe des tonctions

2 + h(x.a) (p(x) - a )

done puisque

p(x) - a

2

•

ilf

ilex feiWf(x.a)da + al(x) +

Le dernier terme peut s'eerire

J

Jeiwf(x.a) a da

e iwf(x.a) h( x.a ) ~ ila d a

162

a

aa h(x,a) da

ioo il est done d'ordre

100 .

En repetant Ie meme raisonnement on trouve un develop-

pement de la forme u(x)

'\,

1:

Jeioof(x.a) da +

oo- q b (x) q

q

r

00-q c

(x)

q

q

_ a(x) + p(x)

- a 3/3

J e ioof(x ,«) ada

Dans Ie cas considere f(x.a)

on introduit la fonction d'Airy 1

.A,(t) :;

2II

+ ...

J- ...

e

itt - t

3

/3 dt

on a alors feioof(X,a) da

:/3

•

eiwa(x)

Jt(00 2 / 3

p(x»

00

et, en derivant sous Ie signe d'integration par rapport

J

e ioof(x,a) a da

1

·213 00

eiwa(x)

.:i' (i/ 3

a

p:

p(x»

D'oii

On

a les estimations suivantes pour la fonction d'Airy

.A: (t)

~

.:/C' (e)

:lL

J

~4 _ t l/4

"n-

2 3/2 cos(} t

114 )

pour t > 0

. 2 3/2 un(} t -

114 )

pour t > 0

grand

.

. grand

(peut etre obtenu par la methode de la phase stationnaire). On

voit que pour p > 0 et

pement :

2 3 / grand on a pour Ie premier terme du develop-

00

163

'V

u ( x) _

L

eiwcr(x) i jj 2/j

vu w

(w

p)

1/4 [b O cos(

2w3P2/3

-1!.4) - Co p I/ 2 Sin(2Wp23/3 - 1!.4

J

cette expression coincide avec celIe obtenue dans les etudes d'optique geometrique en presence de caustique, on cons tate en particulier un changement de phase de n/2 entre les deux termes. Remarque

pour t < 0,

cit (t) it' (t)

'V

'V

Ie]

grand on a -:

un 1

- 2/3 (- t)3/2 1 e(_t)1Y4

un

(_ t) 1/4 a - 2/3 (- t) 3/2

Si l'on admet que l'integrale donne aussi une estimation de u pour p
£ ~~

./1.1-

r

~ t:

che coincidono con Ie (3.9), quindi (3.3) e (3.10) rappresentano una coppia di va10ri di E e H che soddisfano aIle equazioni di Maxvell.

4.

Passiamo ora allo studio delle soluzioni delle equazioni

di Maxve11 trovate nel numero precedente. Supponiarno anzitutto la lamina di spessore infinito, ossia s=

"0

,

sd.cche 1a lwdna occup a i1 semispazio z .s14

~o

•

Supponi arno che per t=t~, z=O'fEocG(to) e proponiamoci di determinar e i v alori di z e t per cui E rimane uguale a Eo, valori che rappresenteranno una caratteristica dell'equazione (3.6), come ha osservato i1 Prof. Jeffrey. Si ha percio l'equazione:

certamente soddis f atta se:

205

Di££ereDZiando s1 ha subitOI (4.3)

Ora dz

~

10 spostamento del campo elettrico di valore

tempo dt. quiJldi

Re-.,)

E~

nel

~ la veloci ta con cui si propaga il

campo elettrico che all'istante to aveva il valore

E••

Notiamo che la (4.2) si puo ricavare in altro modo d1 va11dita

pi~

amp1a. Si osservi in£atti che se G(t - p(E.,)z)

e

costan-

te e uguale a Eo. per la (3.6) s1 hal

da cui integrando e teJlendo conto che per t=t o

,

z.o , s1 r1-

trova (4.2). E' bene nctare che, essendo ~1/~r pos1tiva, 11 campo s1 propaga Bel verso positivo dell'asse z, qu1ndi la soluz1one del numero precedente rappresenta un'onda che s1 propaga nel verso positivo dell'asse z. In particolare, se il campo

~

nullo per z=o

in un certo istan-

te to , esso s1 propaga con veloc1ta l/p(O). Hel caso per noi

pi~

interessante in cui G(t)=O per t ... 0, si

c ompz'ende che per t> 0 si avra un £ronte d' onda, cioe ua piano/ che si sposta col Z

temp~ di

ascissa z 0

..

Zo

(t ) tale che per

:;>zo, E(z,t)::o, per z

~M'

=

~ ..

come si era arPermato. Assumeremo h T

pu~

Per~

~

he> l'estremo inPeriore delJli h(t) per t

essere aache inPinito nel caso G(t)=O per

soddisPatte) e t non

purch~

.T).

sia soddis£atta (5.1).

t~O (sicch~

e molto

t (- 0 0

(5.1) sono certamente

elevato, segue h-

00.

InPatti

sia p~> 0 il minimo valore di peE) per IE\~ M, allora se valgono le relazioai:

(5.4)

t

~

T

t

il valore di u che

(5.5)

u

=t

< P.

Iho..

compar~

- p(E)h(t)

E;

h

0

= r......, NM'

nella (5.2)

~:

t - 'D ...... h < P.-.. h - P-. h

Ma allora il G'(u) della (5.2)

~

= O.

nullo e questa equaziORe nOB

puo essere soddisfatta per h(t) finito. Deve essere h= 00 10. che

~

10 stesso, la soluzione (3.3)

e

valida,per valori di t

soddisfacenti (5.4), per ogni z, ed essa rappresenta il campo elettromagnetico in tutto il semispazio. Si noti chef come vedr~~o ~el

numero seguente, N ~ molto piccOlo; l'intervallo di

tempo in cui la (3.3)

e

valida PUQ essere sufficentemente gran-

de per Ie applicazioni pratiche. Hel caso in cui non siano soddisfatte Ie ipotesi ora esposte. fissato t PUQ esistere un valore:z di z per cui la (3.5)

e

nul-

la, e se G(t - p(E)z) risulta diverso da zero, da (3.6) e (3.7)

I

fl. ' 9E . ". ' l;lE segue che \ ~ ~J: =-t- " 0 , I i-:i I.)f; I = ... c>o , C10e Ie derivate di E per z _z tendono a diventare infinite. Si ha cioe,

conforme a una locuzione del Prof.JefPrey, una catastroPe. Si PUQ cosi interpret are l'accennato risultato di Cesari per cui i Suoi teoremi sono validi solo mina

e

suPPicentemente piccOlo.

~e

10 spessore della la-

208

In seguito cornunque ammetteremo che (3.3) e (3.10) rappresentino il campo elettromagnetico, almeno per valori di t e z sU££icentemente grandi per Ie questioni pratiche.

6.

=

Hel case s

notiamo che, mentre (2.12) rimane valida,

00

(2.14) non ha pi~ signiPicato e si pu~ sostituirla can la con-

dizione che il campo sia nullo all'infinito,

0

meglio che il

campo rappresenti un'onda che si propaga nel verso positivo dell'asse z, condizione questa, come si

~

osservato al •• 4, sod-

dis£atta dalle (3.3) e (3.10). Supponiamo ora l ' onda E
t:L"

senl.Dt

(A 0 e

w costanti)

•

Converra introdurre la Punzione di Heaviside let), (l(t)=l per tpO, l(t) =0 per tC4J] (q~'

1.

N.

=

mo'"C

(1Z. (4)J.

4.6 )3 • Le condizioni di compatibilita che coincide con 2termomeccaniche aono qUindi atte a determinare UN ed inaie=

250

me le discontinuita di

e

e .....

di ~ (I) in funzione di @ •

5i ha anche ( cfr. ( 1.19 )1 )

c~ con

~:

uJ

0

=

U

= un

- vn

velocita di avanzamento ( locale di propagazione )

dell'onda. Linearizzando

(

[~1

=

(~J.

N

= - UN ~o

= - UN a

• N

e. )

e quindi

[~(t)J = - ~o a a @ > l'onda e compresaiva se 4.17 )

e: 0 , espansiva nel

caso

opplilsto (3) (3) Si osserva che ( 4.17 ) vale anche se 1'onda e 1ongitudi= nale, mentre ,per onde meccaniche (e = 0 ) implica La continuita di ~u;

Per onde Longd, tudinali,

UN

non puo essere determinato da1=

le sole condizioni di discontinuita,mentre per le discontinui= ta di JC , di '1.{')

[lel ( 4.18 )

si ha

=

a @ ( ? + 2}J- -

~ -~" a

u; )

251

mentre

e

anche

( 4.19 )

BnlLIOGRAFIA A

Vincoli

1 - J.E.

Adkins

: A three-dimensional problem for

highly

elastic materials subject to constraints,Quart.J.Mech.Appl. Math,11,88-97 ( 1958 ) 2 - J.E. Adkins - R.S.Rivlin

: Large elastic deformation

of

isotropic materials : X • Reinforcement by inextensible cor: ds, Phil.Trans.Roy.Soc.London A, 248 201-223 ( 1955 ) 3 - T.Alts -

Termodynamics of thermoelastic bodies with

kinematic constraints : fiber reinforced materials. Arch. Rat.Mech An. 61 (1976) 253-289. 4 - G.B. Amendola

A genaral theory of thermoelastic solids

restrained by an internal thermomechanical constraints, Atti Sem. Mat. Fis. Univ. Mod. 27 (1978) 1-14. 5 -

J. Bell

: The

experimen~al

nics.Encycl.of Phys.Vol

foundations of solid mecha=

VI a/ 1 ,Springer Verlag ,Berlin

1973. 6 - G.Capriz - P.Podio- Guidugli

Formal structure and

classification of theories of orient ed materials,Ann.mat. pura appl.(4) 115 (1977)17-39. 7 -

H.H. Erbe

W~rmeleiter

mit thermomechanischer in=

neres Zwangsbedingung,ZAM,Sonderheft

76-79 (1975)

252

8 - J.L. ErlksGn- R.S.Rivlin

: Large elastic deformations of

homogeneous anisotropic materials, J.Rat.Mech.Anal.3,281361,(1954) 9 - A.E.Green-J.E.Adkins : Large

elastic deformations . and

non -linear continuum mechanics,· Oxford,Clarendon :eress 1960 ~O-

A.E. Green- P.M.Naghdi-J.A.Trapp : Thermodynamics of a con= ~inuum

with internal constraints,Int.J.Enw1g.

.Sei.8 (1970)

891-908. 11- M.F.Gurtin-P.Podio-Guidugli : The thermodynamics of constrai ned materials, Arch.Rat.Mech An.51 (1973) 192-208. 12 -E. Hellinger , Die al1gemeinen Ansatze der Mechanik der Kon= 4, tinua,Enz.math.Wiss. 4 602-694 (1914). 13 -S.Levoni:Sulla integrazione delle equazioni della termoe1a= sticita di solidi

incomprimibi1i,Ann.Sc.N~rm.Sup.Pisa

(3),

22,(1968),515-525.. 14 -T.Manacorda:Una osservazione sui vincoli interni in un soli= do,Riv.Mat.Univ.Parma (3),3,(1974),169-174. 15 -T.Manacorda:Introduzione al1a

termomeccanic~

dei continui,

Quaderni UMI,n.14,Pitagora Ed.,Bologna 1979. 16 -S.L.Passman:Constraints in Cosserat surface theory,Ist.Lomb. Sc.Lett. Rend. A,106,(1972),781-815. 17 -A.C.Pipkin:Constraints in linearly elastic materials,J.of 6,(1976),179-193. 18 - H.Poincare:Le90ns sur la theorie mathematique de la lumie= El~

re,Paris, 1889. 19 -H.Poincare:Le90ns sur la theorie de l 'elastcite,Paris,1892. 20 -S.Rivlin:Plane strain of a net formed by inextensible cords, J.Rat.Mech.Anal. 4,(1955),951-974. 21 -A.Signorini:Trasformazioni termoelastiche finite,Mem.III, Ann.Mat.pura appl. (4),39,(1955),147-201.

253

22 - A.J.M.Spencer:Finite deformations of an almost incompres= sible elastic solid, Second-order Effect in Elasjicity, Haifa 1962,Oxford,Pergamon Press,1964. 23 - J.A.Trapp:Reinforced materials with thermo-mechanical con= straints,Int.J.Engng.Sci. 9,(1971),757-776. 24 - L.R.G.Treloar:The physics of rubber elasticity,Clarendon Press, Oxtord,1949. 25 - C.Truesdell - W.Noll:The non-linear field theories of Me= chanics,Encycl.of Phys. Vol.IIl/3,Springer Verlag,Berlin, 1965.

1Il

) Omie.

1 - A.Agostinelli:Sulla propagazione di onde termoelastiche in un mezzo omogeneo e isotropo,Lincei Rend. (8),50 (1971)

163-171,304-312. 2 -G.B. Amendola : On the propagation of first order waves in incompressible thermoelastic solids, B.U.M.l. (4) 267-284. 3 -

1

G.B.Amendola: Acceleration waves in incompressible rials. Atti

Sem.Mat.Fis.Modena

24

mate~

(1975) 381-395.

4 - C.E.Beevers : Evolutionary dilational shock waves generalized

(l~13)

in

a

theory of thermoelasticity,Acta Mech. 20(1974)

67-'19. 5 -

a.R.Bland: On shock waves in hyperelastic media,IUTAM, Second order effects in El a s t mc i t y , Pl a s t i c i t y and Fluid Dynamics,Pergamon Press,1964.

6 - P.Chadwick - P.Powdrill:3ingulnr surfaces in_linear thermo=

254

e1asticity,Int.J.Engng Sci. l (1965) 561-596. 7 - P.Chadwick-P.K.Currie : The propagation and growth of acce= 1eration waves in heat-conducting elastic materials, Arch. Rat.Mech.An 8 - P.J.Chen

49

(1972) 137-158.

Growth and decay of waves in solids,Enc.of Phy=

sics, VI a/ 3 Springer Verlag, Berlin, 1973. 9 - P.J .Chen - M.E.B:urtin

: On wave propagation in inextensi=

ble elastic bodies, Int.J.Solids Struct .1Q

(1974) 275-281.

10 - P.Chen-J.W.Nunziato :On wave propagation in perfectly heat conducting inexteneible elastic bodies,J.of El.5(1975) 155-160. 11 - B.D.Coleman, M.E.Gurtin,I.Herrera,C.Truesdell : Wave

prop~

gation in dissipative materials,A reprin' of five memoirs, Springer Verlag,New York 1965. 12 - WgD.Collins : One-dimensional non linear wave propagation in incompressible elastic materials,Q.J.Mech.appl.Math.19 (1966) 259-328. 13 - J •Dunwoody

: One dimensional shock waves in heat conduct!

ng materials with memory : 1- Thermodynamics-Arch.Rat.Mech. An.~7

(1972) 117-148; 2-Shock analysis, ibidem, 192-204;

3- Evolutionary behaviour, ibidem 50,278-289. 14 - J.Dunwoody:On weak shock waves in thermoelastic solids,Q. J.Mech.appl.Math.30 (1977) 203-208. 15 - J.L.Eriksen:On the propagation of waves in isotropic incom= pressible perfectly elastic materials,J.Rat.Mech.An. 2 , (1953),329-338. 16 - A.E.Green-P.M.Naghdi:A derivation of jump condition for entropy in thermomechanics,J.of Elast. 8 (1978) 179-182. 17 - E.Inan:Decay of weak shock waves in hyperelastic solids, Acta Mech. 23 (1975) 103-112.

255

18 - T.Manacorda:On the propagation of discontinuity

wave~

in

thermoelastic incompressible solids,Arch.Mech.Stos.(2), 24 (1971) 277-285. 19 - T.Manacorda: Zagadnienia Elastodynamiki,Ossolineum,Varsa= via,1978~

20 -

Cfr.anche A - 15.

R.W.Ogden:Growt~

and decay of acceleration waves in incom=

pressible elastic solids,Q.J.Mech.appl.Math. 27 (1974) 451-464. 21 - N.Scott:Acceleration waves in constrained elastic materials Arch. Rat.Mech.An. 58 (1975) 57-15. 22 -

N~Scott:Acceleration

waves in incompressible elastic so=

lids,Q.J.Mech.appl.Math. 29 (1976) 295-310. 23 - N.Sc_ott-M.Hayes:Small vibrations of a fiber-reinforced composite,Q.J.Mech.appl.Math. 29 (1916) 467-486. 24 - C.Trimarco:Onde di accelerazione in materiali termoelasti= ci con vincolo di inestendibilita,in pubbl. su Atti Acc. Sci.Modena. 25 - C.Truesdel1:General and exact theory of waves in finite elastic strain,Arch.Rat.Mech.An. 8 (1961),263-296.

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

"ENTROPY PRINCIPLE" AND MAIN FIELD FOR A NON

LINEAR COVARIANT SYSTEM

TOMMASO RUGGERI

INTERNATIONAL MATHEMATICAL SUMMER CENTER (C.I .M.E.) 1" 1980 C.1.M.E. Session: "Wave propagation".

Bressanone 8-17 giugno 1980 . "ENTROPY PRINCIPLE" AND MAIN FIELD FOR A NON LINEAR COVARIANT SYSTEM

by TONNASO RUGGERI

Istituto di Matemat ica Applicata -

Universita di Bologna

Via Vallescura 2 - 40136 Bologna (Italy).

I.Introduction My lecture is complementary to the lectures given by G.Boillat i n the first part of this course. In

part~cular

I am shall deal with some problems concer-

ning quasi-linear hyperbolic system compatible with a supplementary conservation law; relativistic theories will be considered with special emphas i s . I start with a brief bibliographical introduction to the subject I shall be concerned with. In 1970-71 1. MUller, in some works [1] on "rational" thermomechanics of continuous media, proposed the "entropy principle" as a criterion for selecting the constitutive equations. This author considers the equation governing the evolution of a thermomechanic system : a) balance of momentum, b) balance of mass and c) balance of energy equations . Adding the constitutive equations to the prev ious system one gets a system of 5 equations in 5 unknowns. Each solution of this system is called a "thermodynamic process" . Then MUller postu- lates the existence of an additive function cr (entropy) such that : +

d dX i

(PSv. + ~

~ .) ~

= cr

o

lJ

t her modynami c process . (l)

Furthermore he supposes that both the entropy density S and the flux ~i are

....

constitutive functions (p and v are respectively the mass density and the velocity). Hence, from (1) further constrains arise for the constitutive rela-

260 tions,

besides

the

usual

ones

which

can be

imposed

according

to

the

principle of material objectivity. In particular the author shows which

he

identifies

with

the

the existence of a universal function,

absolute

temperature ;

hence,

he

deduces

the first principle of thermodynamics . In a different conte>ct in 1971 K.O.Friedrichs and P.D.Lax [2] and later the

former

in

1974

[3J

examined

a

similar

problem .

In

particular

in

[3] Friedrichs, in a covariant formalism, considers a conservative quasilinear hyperbolic system of r first order equations of the type

aa

~

o

(U)

= .2(U)

(*)

(2)

In (2). N eqs. may be identified with the field equations. while the remaining r - N are supplementary conservation laws. Then

comp~

tibility conditions are required in order that the system has a solution. In particular, when r=N+l (one supplementary law), as the system is quasi-linear, compatibility is ensured by the existence of an r-vector

z(U).

such that: a

= 1-.2 I,l V = a/au we have =0

I·aa~

Introducing the operator (condition I)

U,

I,l

aa u. (3)

Moreover Friedrichs supposes another condition holds: it exist at least a time-like covector

{~

a

}, independent of the field, such that the quadratic

form ( condi tion II) Here

oU

is positive definite .

(4)

is an arbitrary variation of the field and

o2~t

= su - VV ga su .

Using condition I and II Friedrichs shows that the system of the field equations is a hyperbolic symmetric system . (*) To avo id misunderstanding the vectorErn

r

are underlined .

261

Later several authors [4J. [5], [6J provided further contributions on th is subject. especially concerning shock waves in non-covariant formalism. Now we shall obtain the above mentioned results in an explicitly covariant formalism. dealing with

the physically relevant case of one supplementary

law. The covariant formulation allows to apply the results to explicitly covariant theories and. moreover. to emphasize some conceptual aspects that, in our opinion. have not yet been pointed out. 2. Main field and Covariant convex density. Let V~ be a C·, 4-dimensional mani f ol d and x a point of v~. x a being local coordinates of x. The manifold is supposed to endowed with a pseudo-Riemannian a

metric . In the local coordinates x • g

a8

represents the components of the

metric tensor of signature (+ - - -) . On V~ we consider a quasi-linear conservative system of N first order partial differential equations for the unknown N-vector

U(xa)E~N (5)

the components of Fa and U are contravariant tensors and da is intended as a covariant derivative operator. We suppose that the system (5) is hyperbolic. Le. :

.:J

a time-like covector

{Cal. such that the following two statements hold : i)

(6)

ii) V covector{~a} of space-like the eigenvalue problem a (~ a- .llCa ) A - d = 0

(7)

has only real proper solutions ll(k) and a set of linearly independent eigenvectors d The

(k)

(k=I.2 ••••• N). • where II is solution of (7) are called "characteristic". a} fUlfilling L}, ti) are said "subcharacteristic".

covectors{~a

while the{c

a}

- llC

262

When a differentiability conditions holds, let us suppose that (5) is compatible with a supplementary conservation law

aa h

a

a

(8)

h (U) = g(U),

being a contravariant vector and g a covariant scalar. In this case we may write the conditions I and II of Friedrichs in a more

convenient form. We have:

ga -

(J

E - (: )

Since by (3) I is defined up to a scalar factor, we may write

I

='

then Friedrichs conditions lock like U' • VFa

uv-r

=

V h a,

(9)

= g,

(10)

We remark that (9), multiplied by 6 U, can be written equivalently: V

s u.

The identity (12) show the first important result : ,

U' i s invariant with respect to field 0,

i.e . to the convexity of the covariant scalar density h=hal;a with respect to a the field U = F I;a'

If for a system (5) there exists a vector U' and at leaet: a eooeetor (I;a}suah that (12) and (17) ho ld, we say that the system is a aonvex aovariant density system. Conditions (16) and (17) ensure also that the mapping

U'+~U

is globally

invertible , becauseVU' =VVh and th is gradient matrix is symmetric and positive definite; then for a theorem about globally univalence ([8J) U'

+~

U is glo-

N bally univalent in every convex open domain D ~ R Therefore it is possible to choose the vector U· itself as field variables and prove that in this case system (5) has the form L {U'}

where the operator

~

f(U' )

(18)

is given by A' a ~

(t)

(t)

aa

(19)

For the proof of the statements proposed in this lecture one may see [9J •

264 and (20) System (18) is symmetric hyperbolic : in fact a system A,a a U' = f is sima a aT a metric hyperbolic if ~' A' and A' ~ is positive definite, and in our a a case we have A' ~ a ='J' 'J'h', but from (20) h ,a ~ = h' = a = U'·U - h is the Legendre conjugate function of h and then it is a convex ~

function of U' • We remark also that the differential operator in (19) depends only on onefour-vector h ia and this justifies our definition of "four veator generating funa-

tion" for the symmetric system . We have seen that any convex covariant density system is endowed with a vector U' that may be expressed as a function of the field variable and is invariant with respect to transformations of field. In fact it is determined completely

only

by

the

conservative

system

law

(8).

the

system assumes a symmetric form,

is

well

Moreover we pointed out

posed.

Such

remarkable

that, so

(5)

and

when U'

the

supplementary

is chosen as

that the local Cauchy

properties

suggest us

to

call

field, problem

U'

the

"main field" of the system. We remark that not only on the mathematical point of view U' and h,a possess a special role with respect to other quanti ties, but also from the physical point of view, they are privileged, since they are related to the "observables" of the physical system, as we shall see later. System

(5),

compatible with

(8),

is

riducible

to the form

a suitable choice of the field variables and viceversa;

for

(18)

in fact it is

easy r:o prove that the system (18) provides always a supplementary law (8) :

a let h

= U'

• 'J'h'Cl

_ h,a, we have

U'·f =g. Finally we have shown that

a

a

h

Cl

'J'h

Cl

•

ac

U'

U' •

f { U'

}

265

A necessary and sufficient condi t i ons f or the sys t em

( 5)

to be compat i ble

with a supplementary conservation law (8 ) with h ( U) convex fun c tion wi th r es a pect to U = F ( (con vex covariant dens ity systems) , is t hat t here exists a a

choice of the f i e ld

(invariant r espect field transformations and indepen-

U'

dent of t he congru ence defined by the time-like covector {(a })' so t hat t he system (5) as sumes the syrmzetric f orm ( 18) with h'=h, a ( Th is

proposition

is

a

f irs t

c ontr ibut ion

of

the

a

convex fu nc t i on of U' •

que s t l on

proposed

by

LMUller ("a c hallenge to mathemati cians" [lJ ) . At

least we point out t hat if in (8 ) we impo se the c on di t i on g

> 0,

t h e n , by (10) all solutions of (18) satisfy

3 . Shock Waves Theory for Convex Covariant Density Systems.

i) Entropy growth across a shock wave. Let

a

Il

connected

open

se t

i n t o two open subset Il l. Il z. of

r :

we shall

Let ~( x

r

ind en t ify

v"

of

a

and

a hypersu rface cutting Il m (m> 2) , be the equation C

r

) = 0, ~

E

with a shock hypersurf a ce for the fie l d U.

It is known that the Rank ine-Hugon iot c ondit ions mu s t h o ld [ F a]

~

= 0, on

a

r

where brackets denote t he jum p a c ros s Formally

t he

Rank i n e-Hug on io t

r,

(21) = aa ~ .

~a

and

equations

a re

ob t a i n e d

from

the

field

e q s . (5) through the correspond e n ce rule

a

...

a

However

this

r ul e

[ J

~

a not work

does

(22)

wh en

a pplied

to

the

s up pleme n tary

equation (8); in fact

n does

not,

n is

non

in general , negative.

[4J

by P .D.Lax

[h a ] ~

=

van i sh .

Th i s

n for

dyna mic n

>

0

,

on

( 23)

r ,

Fu rthermo r e

r esult wa s

pr oven

1

t

is

in

to

show tha t

a non c ova r i a n t

possib le

f ormal i sm

introducing an ar tifi c al vi scosity in the f i el d e qu ations;

a different proof was given in of

a

[51.

It is know that th e pos i t i v e s i g n a t u r e

t he n on r e l a tiv i stic perfect fluid implies the g r owth of t hermo e ntropy is

a cros s

o f t en

t he

c alle d

in

s h oc k. the

That

is

the

literature

reason why the c ond it i on

" entr op y

gr owth

c ondi tion"

266

a nti i s

assumed as a criterion to pich u;> the physica l

shocks a mong the

solutions of the Rankine-Hugoniot equations. In this section we suggest the main steps of an explicitly covariant proof of the fact that n is non negative on Let U; }

r. 1 and a a

be a subcharacteristic covector such that

a

covariant scalar defined as :

a = -

(24)

f;a4>a

thm there exists a space-like covector {l; } a

such that f; l;

a

a

°

be

= 0.

(25)

the equation of a characteristic hypersurface which

locally has the same "direction of propagation"

l;a

of the shock surface.

i.e. (26)

0,

(27)

where

(k)

(k=1,2 •..•• N) are

IJ

the solution of (26);

these eigenvalues

are real by the hyperbolicity condition. Now

we

consider

a

solution

of

the

Rankine-Hugoniot

equations

(21)

(shock) : (28)

u,

U* being

the

perturbed

and

the

unperturbed

fields

respectively

on

r (in the following * will denote the values of any function of the field

computed

for

U

=

U*) .

Here

we

take

only k-shocks

according to

the following

De f i ni t i on of k-shock

We

shall

say

that

a

shock

is

a

k-shock

if

there exists a number k (=1,2 •••. ,N) such that lim

O-+-\J(k)

*

U = U·

(29)

267

Roughly speaking a k-shock is a shock that vanishes when the shock speed approaches to a characteristic velocity weak shocks when We suppose

0

~~k)

is near to

to know the

(of course,

these shocks become

).

solution

(28)

for

into (23): then we get n as function of u* and

a

k-shock

and

replace

it

~a

(30) By differentiating

(30)

respect

to

~a

and

taking into account

(9),

after some calculations we obtain

[h

a

J-

Vhf

a

[F ]

Thus (31) Since h is a convex function of U, defined in a convex domain D, we have: w(U,U*) = -h(U) + h(U*) + Vh'(U - U.. ) > 0,

So the r.h.s. in (31) is equal to

-\!T,

F;a an/a~a

Furthermore,


0

is

a

p

smaller than the velocity of

light in vacuum. Hence

the

system

of

hydrodynamics

equat.ions

is a

particular case of

the general theory. As a consequence we have:

The system of re 1Ativistia hydrodynamics equations is syrrunetric hyperbo ldc

1)

in the fieUl. U' (44) with the four-veator generatirlfJ functions simpLy given by

This is a

r,

on

2) [ 5 ] > 0

when

conseq~ence

of i) since it is possible to show that

a n = _r*u ~

* a

where

)1*

is a

a

[5]

= r*(o_)l*)u

(

* a

[5] •

(46)

characteristic speed of the corresponding material wave: IJ*

= (u o *

1;) / (u 0

a( *

) •

a

3) The knauledqe of [5 ] as a function of u* and ~ 0 detierminee the shook, This is a consequence of ii) and (46).

The veLocity of propagat.ion of the reZativistia hydrodynamias shocks never

4)

exceeds the speed og Light. REMARK

We have seen above that the main field U' of a convex covariant density system is may

be

invariant

expressed

as

under

transformations

of

the

field

U.

and that it

the gradient of the convex covariant density h = a = hat • with respect to the field U = F (0 here h represents the a a component of the proper density of the conserved quantity h in the

272

congruence defined by the time-like covector It

is

another {u} a

remarkable convex

that,

density

in the fluid

h,

relative

to

}. a case, it is possible to define {~

the

field-dependent

congruence

with the same properties of h

h = ~u - rS , a whose gradient with respect to the field

(:' ) is still equal to the same main field U' (44)

u'

dh

Le.

Vh, (V -d{rS)

=

waul

= -(ua/e)d(pu a)

+ {f/e-S)dr

as it immediately verified taking into account (41) and (42). Moreover it is possible to prove that wi th respect to (il pi ilp)S ~ 1.

U and

h = -rS

is still a convex function,

the prove does not require the auxiliary condition

Hence the mapping U' ..... U is globally univalent.

REF ERE N C E 5 [lJ I.MULLER, Habilitationsschrift an der RWTH Aachen (1970). Arch.Rat.Mech. Anal. 40, 1-36 (1971). (See also "Entropy in non-equilibrium - a challenge to mathematicians" in Trend in Ap~lication of Pure Ma~1ematics to ;.Iechanics, Vol.II; Ed. H.Zorski, Pitman London, 281-295 (1979)). and

P .D.LAX,

Proc.Nat.Acad.Sci.U.S.A.

68

1686-1688

[2J

K.O.FRIEDRICHS (1971) •

(,]

K.O.FRIEDRICHS, Comm. Pure Appl. Math. 27, also Comm. Pure App1. Math. 31, 123-131 (1978)~

[4J

P.D .LAX, ~Shock waves and entropy" in Contribution to non linear functional analysis Ed. E.H .Zarantonello, 603-634 New York; Academic Press (1971).

749-808

(1974).

[5) G.BOILLAT, C.R. Acad.Sc. Paris 283A, 409-412 (1976).

[6J

G.BOILLAT and T.RUGGERI, C.R. Acad.Sc. Paris 289A, 257-258 (1979).

[7] I-SHIH LIU, Arch. Rat. Mech. Anal., 46, 131-148 (1972) .

(See

273

(8J

M.BERGER and M.BERGER , Perspectives Inc. New York, p.137 (1968).

in

nonlinearity . . W.A.Benjamin;,

[9]

T.RUGGERI and A.STRUMIA, "Main field and convex for quasi-linear hyperbolic systems. Relativistic (to appear).

densi ty dynamics".

covar~ant

fluid

CENTRO INTERNAZIONALE MATEMATICO ESTIVO (C.I.M.E.)

SINGULAR SURFACES IN DIPOLAR MATERIALS AND POSSIBLE CONSEQUENCES FOR CONTINUUM MECHANICS

B. STRAUGHAN

CIME Session on Wave Propagation Bressanone, June 1980

Department of Mathematics, University of Glasgow University Gardens, Glasgow G12 8QW

Singular surfaces "i n dipolar materials and possible consequences for continuum mechanics. B. Straughan, University of Glasgow.

1.

Introduction In this paper we study the evolutionary behaviour of a

propagating singular surface in two types of nonlinear dipolar materials;

a compressible inviscid dipolar fluid and an elastic

dipolar solid. The basic theory we use was introduced by Green and Rivlin

[1] and from the constitutive theory viewpoint essentially extends classical continuum mechanics by including gradients of the independent variables present in non-polar theories.

Gradient type

theories were suggested earlier by, for example, !1axwell and by Korteweg, see Truesdell and Noll [2], §125;

in particular,

Korteweg developed an interesting theory of surface tension by allowing for the possibility of rupidly changing density gradients in an interface.

Since in a singular surface quantities such as

density and its gradients of various orders may change very rapidly a study of wave motion in multipolar materials may prove of value. for an elastic dipolar material the theory we empl oy was derived by Green and Rivlin

[1], whereas the

con s t it ut iv ~ develop-

ment for dipolar fluid theory is due to Bl eus t ei n and Green [3]

278

(later modi f i ed by Green and Naghdi [4J). This theory allows for an additional dipolar stress as well as the normal one . In contrast with Newtonian theory the constitutive variables include temperature. velocity and density gradients. This is in one sense a generalization of the Maxwellian fluid of Truesdell (see [2]. §125 and the references therein) in that a dipolar stress is included from the outset. although Truesdell's Maxwellian fluid involves a constitutive theory which includes density. temperature and velocity gradients of arbitrary orders. We pay particular attention to the compressible dipolar fluid since as Truesdell and lIoll [2J point out. the Maxwel lian theory.

• •• "is set up in such a

~Jay

as to emphasize effects of

compressibility" • The dipolar fluid equations given by Bleustein and Green The equations of morner.tum and continuity

[3] are now reviewed. are (1.1)

0 • •• J1-~J

+ pf., 1-

where standard notation (see ego [3)

is employed throughout.

However, the energy equation takes form (1.3)

PI' -

peA

+

70s

i-

TS) - q • . + 1-~1-

.. + r.(")kA~ .. J1-d1-J 1-J :;)1.

t ..

= 0,

where Akj' = 1Jk •• and T •• and r. i j k are a symmetrr-Lc tensor and 1~J1J1the dipolar stress tensor, respectively, related by the equa t Io-.

(1.4)

'ij

= °ij

+ r.kij~k + p(Fi j -

ri j

)·

Here Fi j are components of a dipolar hody force and

ri j

is th e

dipolar inertia which has form (see Green and Naghdi [4])

(1.5)

rj i = d2(Vi~j

- 1Ji~k1Jk~j - 1Ji~k1Jj~k + 1Jk~j1Jk~i)'

d 2 being the inertia coefficient.

Furthermore, the entropy

inequality takes the form (l.6)

- peA· + S,,-) l'

q.T .

.:!:.-t..:!:.. T + 'ji d ij + r. (ij)kAkji -> 0 •

279

Bleustein and Green [3] develop the equations for a compressible, viscous dipolar fluid.

However, as we are

primarily interested in wave motion we shall derive the constitutive equations for a compressible inviscid dipolar fluid, following the procedure given by Bleustein and Green. Suppose then, that A, S, qi' variables

T

i j and L(ij)k depend on the

(1.7)

Using (3.4) of Bleustein and Green [3], the entropy inequality (1.6) may be written as aA

(1.8)

_.-

p--T . . aT •• ,tJ ,tJ

t

p

q.T . -

tT,t

t

T.-do. Jt 1-J

L( •• )kAY,.,. . ~ 0,

t

tJ

, "-