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oo) : R(x,x,C) = E~-RkC _ f e + * w i t h l o c a l coefficients Rk, which means that they are polynomials in u,u',u", The integrals of these coefficients J Rkdx are first integrals of the KdV-equation. As a matter of fact, instead of the resolvent, another function, the socalled "zeta-function of the operator L" can be considered. This is t r L _ s . It is connected with the resolvent by the Mellin transformation. The above
4
Soliton Equations
and Hamiltonian
Systems
coefficients Rk are equal to the residues of the diagonal of the kernel of the operator L~s in the points s = —1/2,1/2,3/2, An explicit expression for the coefficients Rk can be obtained as follows. The Green function R(x, y, () is a product of two solutions of the equation (L + Qtp = 0 : R(x, y, £) = ip(x, CWiv, 0- ^ i s e a s y *° s e e * n a t R(x,x, Cj = ip(x, ()ip(x, £) satisfies the third-order equation R'" + 4uR' + 2u'R + 4C-R' = 0.
(0.6)
After multiplication by R it can be integrated once 2RR" - R'2 + 4(u + C)^ 2 = c(C),
(0.7)
where c(£) is an arbitrary constant formal series. Taking c(£) = 1 and substituting the formal series R = (1/2) £ £ ° RkC~k~1/2 into Eq. (0.7), one obtains a recurrence formula which yields in succession all Rk, e.g. RQ = 1, R\ = —u/2, i?2 = (3w2 + «")/8. (This procedure is described in detail in Chap. 12.a) A problem arises as to how to find all the operators P for which Lax equation (0.5) can be written. This means the following. The left-hand side involves a zeroth-order operator (namely, that of multiplication by the function u). Hence the right-hand side must also be a zeroth-order operator. Thus, all possible operators P must be found such that [P, L] are zerothorder operators. Such two operators P and L are said to form a P-L Lax pair. (More often it is called L-A pair but we prefer to call it P-L pair in honor of P. Lax.) There are various methods to construct the P-operators. One can use the same resolvent. Let us take the first-order operator P = -Rd/dx + R! 12. It is easy to see that [P, L] = 2R'(L + (), whence [P(L + C) _ 1 ,L] = 2R'. The right-hand side is a zeroth-order operator which can be expanded in C~ 1/2 - The operator P(L + C)" 1 = P E ^ ( - l ) f c L f c C _ f c _ 1 can also be expanded in C - 1 '' 2 ; the coefficients being differential operators of growing orders. Thus, P(L + £ ) _ 1 is a generator for P-operators. This generator was found in [DMN]. The Lax equations are u = 2R'k .
(0.8)
If k = 2, this is the KdV-equation. 0.9. We have demonstrated here only one example: the KdV-hierarchy of equations. More general hierarchies are obtained for operators L of a
I n the 2nd edition this is Sec. 3.7.
Introduction
to the First
Edition
5
arbitrary orders, e.g. L = (d/dx)3 + u\d/dx + u0, etc. They are called generalized KdV-hierarchies. b Besides, matrix equations will be considered with L = —d/dx -1 + 17 and also some others. All the generalized KdVhierarchies can be unified into one Kadomtsev-Petviashvili (KP)-hierarchy. 0.10. As it can be seen, in the above discussion all the operators, strictly speaking, were not genuine ones: they did not act as operators in any spaces. Accordingly, neither classes of functions nor boundary conditions were involved. In fact, only the algebra of commutation relations between operators was significant. This gives rise to one feature of this book: the almost complete absence of mathematical analysis in its classical sense (except some facts about Riemann surfaces), but solely differential algebra. There are no convergence considerations, all the series are formal. Another feature of the book is a regular usage of the resolvent (or, equivalently, of fractional powers of operators). We attach also great importance to the Hamiltonian structure of the equations. For the KdV-equation this means a possibility to write it in the form
where H = J hdx is a functional called the Hamiltonian (which is also a local one, i.e. h is a polynomial in u,u',u",...), 5/5u is the operator of variational derivative. The resolvent R(C) has the property S(J Rdx)/5u = dR/dQ, the explanation of which is, in the end, that (L + £ ) - 1 = (d2/dx2 + u + C ) - 1 depends on the sum u + £ and the differentiation with respect to C yields the same as variation with respect to u. Therefore Rk = c5(f Rk+idx)/5u, where c = const., and Eq. (0.8) has the required form. It is easy to verify for Eq. (0.11) that if F = J fdx is any functional, then its derivative with respect to t by virtue of Eq. (0.11) is dF/dt = {H,F}, where {H,F} = f(6H/5u)'{5F/5u)dx. This last expression has all the properties of a commutator and is called the Poisson bracket. For the first time this bracket was obtained by Gardner [Gar], and Zakharov and Faddeev [ZF]. In all the cases besides KdV it is also possible to construct a relevant Hamiltonian structure. For this, a formal algebraic definition will also be given. Following the tradition established in the literature, they are called GD-hierarchies or n t h reductions of the K P hierarchy in the 2nd edition of this book.
6
Soliton Equations and Hamiltonian
Systems
0.12. An important part will be played by the study of the stationary (independent of t) solutions which satisfy stationary equations [P, L] = 0. The significance of these equations was firstly emphasized by Novikov. They are ordinary differential equations; hence the manifolds of their solutions are finite-dimensional. It is remarkable that if one of the solutions is taken as an initial condition for a non-stationary equation of the same hierarchy then at each moment t it remains to be a solution of the stationary equation. Thus, finite-dimensional invariant submanifolds in infinite-dimensional phase space of non-stationary equations can be obtained. This yields finite-dimensional classes of solutions, soliton solutions and also algebraic-geometrical ones. Stationary equations are also of the Hamiltonian type. They have sufficiently many first integrals in involution to be integrable in quadratures, according to the Liouville theorem. We perform this procedure of integrating explicitly. 0.13. The last part of this book is devoted to the so-called multi-time (or the field) formalism. The matter is that sometimes variables are equal in rights, and it is unnatural to choose one of them as a time variable. We construct an algebraic variant of the multi-time variational and Hamiltonian formalism and apply it to our equations. 0.14. We have tried to make this book available to beginners in this area having only basic training in algebra and analysis. All explanations are detailed. A few computations are left to the reader as exercises, which are actually not too numerous. This branch of science is attractive for the author because it is one of those which revive the interest to the base of mathematics: a beautiful formula.
Chapter 1
Integrable Systems Generated by Linear Differential n t h Order Operators
1.1
Differential Algebra A.
1.1.1. Let L = dn + u „ _ 2 d " - 2 + • • • + M0,
d = d/dx
(1.1.2)
be a linear differential operator. Further we shall associate with this operator nonlinear differential equations. The coefficients of these equations will be expressed in terms of polynomials with real or complex coefficients in Uo, ••-,Un-2 and their derivatives of arbitrary orders (i.e. differential polynomials in {UJ}, i = 0 , . . . , n — 2). As far as we discuss construction of these nonlinear equations (and not their solution), the class of functions {ui(x)} under consideration is not important. Therefore we can deal with the differential algebra Au (or simply A) of polynomials of formal symbols {*4 }, where the operator d (a differentiation) acts according to the rules d(fg) = (df)g + f(dg),du 0, j < 0, i + j > - 1 . Then ,
.
x
t+j+i
Check it!
•
1.3.6. Corollary. I res XYdo: = / res
YXdx.
1.3.7. Notice one useful transformation: / res X+Ydx
= j res X+Y-dx
= / res
XY-dx.
1.3.8. Proposition. If X = 52™oo -^i^ 1 a n d -^m = li then the unique \I>DO X - 1 and the unique X 1 /" 1 starting from d exist. They commute with X. Proof. Let X - 1 = d~m + y _ r o _ 1 5 - m - 1 + Y L m - 2 d " m ~ 2 + • • • with indefinite coefficients. X X - 1 = 1 gives
1 + (X m _! + Y-m-X)d~l + (X m _ 2 + Xm-XY-m-x
+ y_ m _2 + rnY'_m_x)d-2
+ ••• = !.
We obtain a sequence of recurrence equations of the form Y_ m _j. = —X-m-k + Qk where Qk are differential polynomials in {Xj} and {Yj}, (j > —m — k). In the same manner one can construct X1/™1 : (X 1 /" 1 )" 1 = X. Further, [X, X" 1 ] = 1 - 1 = 0. From X = X1/™ . . . X1'™ one obtains, commuting both sides with X 1 /" 1 : [X, Xl/m]
= \Xl'm, xl/m]Xl/m
• • • Xllm
+ x 1 / m [ x 1 / m , x1/m]x1/m
• • • x 1 / m + • • • = o.
1.3.9. Corollary. For an arbitrary integer p the operator Xplm can be constructed which commutes with X and whose highest term is dp. Note that X p / m and X « / m commute since X p / m = (X^m)^q. 1 If X = Y^o~ Xid € Rn-i then dx will symbolize the derivation (1.1.4) corresponding to the set of coefficients X = (XQ, . . . , X n _2) (the same letter
n
12
Soliton Equations
and Hamiltonian
Systems
X denotes here a differential operator and the set of its coefficients, this does not bring about any confusion). Let n-2
/
fdx, Sf/SL = ^2d-i-18f/Sui
R-/d~n+1R.
€
o
1.3.10. Proposition. If X = Y,o~2 xiQi dxf
e R
= f res(X • 5f/SL)dx
n-\
then
= (X,
Sf/SL).
Proof. This immediately follows from 1.2.2. 1.3.11. Lemma. The relation d\L
•
= X holds.
Proof. n-2
n-2
dxL = J2 dxuid1 = J2^idi = X.
1.4
•
Lax Pairs. G D Hierarchies of Equations
1.4.1. Return to the operator L (1.1.2). Let Pm = (I™/") + (we shall simply write L™'n). 1.4.2. Proposition. The commutator [Pm,L] belongs to
Rn-\.
Proof. We have [Pm, L\ = [Lm'n
- L™/n, L\ = -{L™/n,
L].
The operator on the left-hand side is differential. On the right-hand side we have the commutator of two operators of the orders —1 and n. Its order is equal to or less than — 1 + n— 1 =n — 2. • We say that the differential operator P (whose coefficients belong to A) together with L make up a Lax pair (PL-pair) a if [P,L] G Rn-ia
PL stands here for P. Lax.
Integrable Systems
Generated by Linear Differential nth Order Operators
13
Thus, for any integer m > 0 we constructed an operator Pm such that (Pm,L) is a Lax pair. Since [P m ,L] € Rn-i, the derivation c?[pm>£,] makes sense. According to 1.3.10, d[PmtL]L = [Pm,L]. Now, let all the coefficients Ui depend on an additional parameter £ m . We can write a differential equation dmL = d[PmtL]L,
dm =
d/dtm
or dmL={Pm,L}.
(1.4.3)
This is equivalent to a system of differential equations on {ui} where i = 0 , . . . , n — 2. The system is determined by two integers, n and m. 1.4.4. Definition. The set of equations with a fixed n and various m is called the nth Gelfand-Dickey (GD) hierarchy. If n = 2, this is the Korteweg-de Vries {KdV) hierarchy. We shall see that all the Eqs. (1.4.3) with a fixed n and various m are compatible, i.e. can be solved together, and a solution depending on all tm (or finite number of them) can be found. More than that, all these hierarchies are restrictions of one big hierarchy (KP) whose equations are all compatible. 1.4.5. Exercise. Let n = 2, L = d2 + u. Find P% and the corresponding equation (1.4.3). Answer. P 3 = d 3 + (3/2)ud+(3/4)u' = d3 + (3/4)(du + ud). The equation is 4wt = v!" + 6uu',
(t = t3)
(1-4.6)
which is the Korteweg-de Vries equation (KdV). 1.4.7. Exercise. Let n = 3, L = d3 + ud + v. Find Pi and write the corresponding equation. Answer. L1'3 = d+ ( l / 3 ) u 9 _ 1 + 0(d~2), is ut = -u" + 2v',
P2 = d2 + (2/3)u. The equation
vt = v" - (2/3)u'" - ( 2 / 3 ) W ,
t = t2 .
Eliminating v we obtain Utt
= -iu(4)_l(W)'.
This is the Boussinesq equation.
(1.4.8)
14
1.5
Soliton Equations and Hamiltonian
Systems
First Integrals (Constants of Motion)
1.5.1. Definition. The first integral is a functional / = J fdx which is conserved by virtue of Eq. (1.4.3), i.e. 0 = dmf = / dmfdx
= / res [Pm,
L]Sf/SLdx.
1.5.2. Lemma. For any k, by virtue of Eq. (1.4.3), dmLk'n
=
[Pm,Lkln}.
Proof. Let k = 1. Then L1/™ = d+v-id~l-\ , all the Vi being differential polynomials in {it,} and vice verse. If one defines dmVi arbitrarily and then shows that dmUi happen to be correct (including dmVi = 0 if i < 0 and i > n — 2), then it means that dmvi were guessed correctly. Let this guess be dmLlln = [Pm,Ll/n] and check it: from L = (L 1 /")" it follows that n-l
dmL = Y,(L1/n)i[Pm,L1/nKL1/n)n~i~1
= [Pm,{Ll/n)n]
= [Pm,L]
i=0
as required. The guess is correct. After that the equality dmLkln = [Pm,Lk/n] when k > 0 can be proved by the same computation. For negative k the assertion follows from the fact that both dm and the commutator are derivations in R: dmL~kln = -L~klndm(Lkln)L~kln and [Pm, L~k/n] = -L-k'n[Pm, Lk/n]L-k/n. • 1.5.3. Proposition. The functionals J f e = I vesLklndx,
fc
= l,2,...
are first integrals of all the equations of the nth hierarchy. Proof. dmJk = I res dmLk/ndx
= f res{Pm, Lk/n]dx
by virtue of 1.3.5.
= 0 •
Note that if k is a multiple of n the first integral degenerates, Jfc„ = 0. Thus, the first remarkable property of Eqs. (1.4.3) is proven: the existence of infinitely many first integrals. 1.5.4. Exercise. Write the three first integrals if n = 2.
Integrable Systems Generated by Linear Differential nth Order Operators
Answer. Ji = Judx,
1.6
Ji = Ju2dx,
15
J 3 = J[2u3 — (u')2]dx.
Compatibility of t h e E q u a t i o n s of a Hierarchy
1.6.1. The second remarkable property of Eqs. (1.4.3) is their compatibility for different m. This means that two or more equations can be solved together, i.e. one can find functions Ui of two or more variables tm satisfying corresponding equations (1.4.3) with respect to each variable. For this vector fields di and dm must commute for all I and m. 1.6.2. Lemma. The equality diPm-dmPi
= [PuPm}
holds. Proof. Using Lemma 1.5.2, we have (the subscript + or — in a circle: ® or 0 , means that this subscript can be skipped) diL™/n - dmLlln
= [Llln,Lm'n\+ TTl/n
Tm/n-i
— L-k+ ,L>+ n
- [L™ / n ,!//"]+ rrV™
r
m
/n]
J© + [Lm ,L_ /n
ln
mn
= [L% , L™ ] + [L ' , L ' \+
rTm/n
]+ - [L,+ l n
r
i/ni
,L'
j+ /n
= [L l , L™ ].
D
1.6.3. Proposition. Vector fields di and dm commute. Proof. Since vector fields are derivations on A, it suffices to prove their commutativity on the generators, i.e. in their action on L. dtdmL - dmdiL = dt[L™/n, L] - dm[Llln, L] = [^L™ /n - dmLl(n, L) + [L™/n, dtL] - [Llln, dmL] = \[Llln, L™/n], L] + [L™/n, [Llln, L\] - [Llln, [L™/n, L]] = 0 according to the Jacobi identity.
•
1.6.4. Remark. Notice that hj_n = d. Therefore, d\L = dL. The action of derivations d\ and d on generators of the differential algebra A do not differ. This implies that d\ = d. One can either identify t\ with x or consider them as different but being involved in all functions only in the combination x +1\.
16
1.7
Soliton Equations and Hamiltonian Systems
Soliton Solutions
1.7.1. The third property of Eqs. (1.4.3) is that they possess infinitely many exact analytical solutions. The simplest are soliton-type or determinant solutions. We will construct a differential operator L whose coefficients are genuine functions of variables x = tj,t2,t3,... (finite number of variables) which satisfies Eqs. (1.4.3) for m corresponding to the involved tm. Let N be an arbitrary natural number (called soliton number, or number of solitons). Let Vk
= ex.V^akntm
+ ak^emakntm,
k =
l,...,N
where {ak}, {flfe}, k — 1 , . . . , N are complex numbers, ctfc ^ a; when k ^ I, en = 1. Let Vi
1 $ = A
I/i
•
VN
1
VN
d (1.7.2)
(JV-l) (N)
Vi
(N-l) • •
gN-1
VN
(N)
QN
VN
where A is the Wronskian of j / i , . . . , J/JV • In the expansion of the determinant in the elements of the last column, d% must be written to the right of the minors. This $ is a monic Nth order differential operator. 1.7.3. Lemma. The functions yk have the properties dmyk = dmyk,
dnyk = ankyk,
$yk=0.
The proof is obvious. Now we construct the operator L by "dressing" the operator dn with the help of the operator 3>: L = $9"$_1.
(1.7.4)
1.7.5. Proposition, (i) The \tDO L is, in fact, a differential operator with the leading term dn, (ii) L satisfies Eqs. (1.4.3). Proof, (i) Rewrite (1.7.4) as L =
($d-I,)dn($d-N)-1.
(1.7.6)
Integrable Systems
Generated by Linear Differential nth Order Operators
17
The operator $d~N is a monk SPDO of order 0, so is its inverse. Then L = YjT^ooui^i where un = 1- Now, L = L+ + L-. Equation (1.7.4) can be written as L+$ - $dn = - L _ $ . The right-hand side is an operator of order < N hence so is the left-hand side, but this operator is obviously differential. It has the property ( £ + $ $dn)Vk = 0 (see 1.7.3). If an operator of order < N sends to zero N linearly independent functions, it is identically zero, L + $ —$0™ = 0. Then L _ $ = 0 and L_ = 0 since $ is invertible. Thus, L = L+, a differential operator. (ii) Equation (1.7.4) implies L l/n
=
$£$-1
)
Lm/n
=
^Qm^-l ^
£«/"$
_ $gm
=
_£«/"$ .
The right-hand side of the last equality is of order < N, and the left-hand side is a differential operator, say Q. So ord Q < N. Applying dm to $2/fc = 0 and taking into account 1.7.3, we have 0 = {dm$)yk
+ $dmyk
= {dm$)yk
+ L™/n$yk
- Qyk
= ((d m $) - Q)yk • The differential operator (dm$) — Q of order < n annuls N functions yk. Thus, ( d m $ ) - Q = 0 and dm$ = L + / n $ - $
= (LX)+L - L(XL)+ = -(LX)-L
A^(X) + L{XL)_
e Rn ,
where z is a fixed real or complex number is called the Adler mapping. Thus, there is a family of mappings labeled by the parameter z. In what follows, the coefficients of the operator X belong usually to A. However, everything in this section is also true in the case when coefficients are in B. 1.8.7. Exercise. Why the mapping A^ defined by the above formula on R~ is, in fact, defined on R-/d~nRand maps it to i?„? A^ depends on zn linearly. We write A^Z\X)
= And"-1 hence X'jin = 0 and XjltTl = 0 since Xjlin € AQ. This contradicts the assumption. • We can make the same remark about the case of B as before. Now we are in a position to prove 1.8.9. We have seen that an element of the kernel of the Adler mapping is uniquely determined by the set of constants in its coefficients. Therefore, it remains to show that a linear combination £) c£Te can have any set of constants in its coefficients. The ^ D O Lr'n is a sum of dr and an operator whose coefficients do not contain constants. Hence Te considered as an element of R-/d~nR-{{z~1)) is i
-
1
Tt = - Y^ (tz)~r~ndr + T* n
r=—n
where T* is an operator without constants in its coefficients. It is convenient to pass from {Te} to another basis. Let eo be a primitive root. All the roots aree = eg,fc = 0 , . . . , n - l . Let Si = Y2Zo 4 * ^ * . * = °> • • • > n ~ l- All the T€ can be expressed in terms of Si. We have Si = z-ldl~n
+ S? ,
Si are operators without constants in their coefficients. It is easy to see that the linear combinations X = Yl?=o ci{z)Si where c;(z) = ^ ^ ° QkZ~k ensure any set of constants in the coefficients Xji. • It is easy to see that the same reasoning works also in case of B which means that no new resolvents appears even if one admits a wider class of operators. If T = YlTjZ~j i s a resolvent, then Tj satisfy Eq. (1.8.12). 1.8.13. Exercise. For n = 2, L = d2 + u, obtain the set of equations for the coefficients of the resolvent T = Sxd'1 + S2d~2 (using H(T) = 0). Answer. S"' + 4uS[ + 2u'S1 - 4z2S[ = 0, S'2 =
-{l/2)S'{.
(1.8.14)
1.8.15. Remark to bibliography. The idea to generalize the KdV hierarchy passing from the second-order linear differential operator to the nth-order operator was suggested by Krichever [Kri76] and Gelfand and
22
Soliton Equations and Hamiltonian
Systems
Dickey [GD76(b)]. The construction of these generalized hierarchies based on fractional powers of the operator L, which is standard now, first appeared in [GD76(a)] for n = 2 and in [GD76(b)] for the general case. The exposition in the latter paper was rather cumbersome (partly because of the unfortunate idea of the authors that all the coefficients in the expression of a power of an operator should be found explicitly). In the same paper the first integrals were found and, virtually, all the ± and res technique was developed (though the notation res was introduced later by Adler). The transparency of presentation that we enjoy today was achieved after works by Manin [Mani78(b)], Lebedev and Manin [LM], Adler [Ad79] and Wilson [Wil79]. In particular, in [Ad79] Adler suggested his beautiful mapping. As to the soliton (determinant) solution we hesitate to say who is the author of the exposed method. We took it from the "mathematical folklore". More precisely, we extracted it from Manin [Mani78(a)] who seemingly attributed it to Drinfeld [Dr] and Krichever.
Chapter 2
Hamiltonian Structures
2.1
Finite-Dimensional Case
2.1.1. To make the book self-contained we permit ourselves to give some well-known facts about Hamiltonian structures. (An excellent account on this can be found in Arnold [Arn74].) The usual classical mechanical definition of the Hamiltonian system is the following. There is a special, "canonical", set of independent variables consisting of two groups, {q1} the coordinates and {p1} the momenta, i = l , . . . , n . There is also a function W(q,p), the Hamiltonian. Canonical Hamilton equations are qi = dn/dpi,
pi = -dn/dqi.
(2.1.2)
This definition depends on special variables. Not every change of variables preserves the form of these equations. The changes having this property are called canonical (it is not an exact definition but here it does not matter). We shall define Hamiltonian systems in an invariant way, independent of the coordinate systems. We expect from the reader only knowledge of the first notions of the analysis on manifolds (see e.g. [BC]): the manifold, the tangent space, vector fields as linear differential operators £ G TM which can act on functions, i.e. £/. Vector fields form a Lie algebra with respect to the commutator [£, 77] = £r) — r)£. Further, we use the cotangent space at the point x e M, T*M. The coupling between the elements £ e TXM and u £ T*M is denoted as ((,u). We shall also use the following: differential forms, inner product of a vector and a form (if £ is a vector and u> an n-form, i(£)u> is an (n - l)-form defined as V £ i , . . . , f n _ i e TxM(i(£)bj)(£i,... ,£„_i) = w (£)£i! • • • ;£n-i))We need also a coordinateless formula for the 23
24
Soliton Equations
and Hamiltonian
Systems
differential dw (formula Lee, see [BC]): V6.---.£n+i(du>)(£i,---,£n+i)
i
+ X J ( - l ) < + M K i . 6-1. 6 . • • • , & , . . . , 4". •••^n+l),
(2-1.3)
i<j
where the caret over a letter (&) means that the variable must be skipped. 2.1.4. Remark. At first sight it is not at all clear that this formula defines dw as a differential form. The value of a differential form at a point must depend only on the values of the vector fields at that same point and not at other points of its vicinity, which is not obvious in (2.1.3). In actual fact (see [BC]), however, everything is right here. The Lie derivative L^ of a form LJ along a vector field £ is L^ui = (di(£) + i(£)d)uj. In the particular case of 0-forms, i.e. functions / , we have L^f = i(Qdf = 2.1.5. Proposition. Then
Let w be a r-form and £, £ i , . . . , £ r
De
vector fields.
r
£ M 6 , • • •, 60) = (i«w)(6 ,...,&•) + $>(&'•••>&&]'•••>&•)• Proof. We have (Lcw)(6, ...,&•) = (»(0<M(6, ...,&) + (*(0--->&>---.£r) + £ ( - l ) < + M & &, & U l , • • • , L • • • . I,", • • • , tr) r = £w(fc, ...,&•) + J ^ - l j M f c , 6], 6 , • • • - L • • • , £r) which is equivalent to the required equality.
•
2.1.6. Corollary. If £, rj G T M and w G T*M, then 7j<W, 0 = , 0 = w(£)) • A system of ordinary differential equations x(t) = t{x(t)) corresponds to each vector field £(x); here a; is a point of the manifold, x(t) € TM, (x(t)f = {d/dr)j{x{t + r ) ) | r = 0 ) Every nondegenerate two-form permits to identify the tangent and the cotangent spaces, TXM -> T^M: f e
rQM ^ -i(f)w G T ; M
(2.1.7)
(the non-essential minus sign stands here for convenience of what follows). In the coordinate form the operation i(£)u) is the lowering of the superscripts with the help of the tensor wy, where UJ = ^w^dx1 A dx?. The mapping (2.1.7) is a bijection. The inverse mapping will be denoted as H : a G T*M H-> £ € T^M, a = —i(£)a> (in the coordinate form this is the lifting of subscripts with the help of the tensor w1*, the matrix (o/y') being inverse to (wy)). 2.1.8. Definition. A nondegenerate and closed (rfw = 0) form w is said to be symplectic. These forms can exist only on even-dimensional manifolds. A manifold with a given symplectic form will be called a phase space. A vector field £ which preserves the symplectic form u>, i.e. L^w = 0, is called a Hamiltonian field. For such a field 0 = L(u = {di{£) + t(0d)w =
d(i(£)u).
26
Soliton Equations
and Hamiltonian
Systems
According to the Poincare lemma a function H(x) exists (at least locally) such that dH(x) = -i{i)u).
(2.1.9)
The function "H(x) is the Hamiltonian of the system. Conversely, to every function %(x) we can find a Hamiltonian vector field £n whose Hamiltonian is %, i.e. dH = —i(£^)u;, since the mapping (2.1.7) is a bijection. The field is Hamiltonian because Liuuj = d{i{iH)uj) = -d{dU) = 0. A differential equation which corresponds to a Hamiltonian vector field x = tn{x)
(2.1.10)
is called a Hamilton equation. 2.1.11. Exercise. Show that the Hamilton equation corresponding to the form UJ = ^2dpl A dq% and the Hamiltonian %{p,q) is Eq. (2.1.2). 2.1.12. Lemma. If h is a Hamiltonian, i.e. a function, and £ is a vector field, then £/i = w(£,&).
(2.1.13)
Proof. £h = i(£)dh = -»(0*(&)w = - " ( & , £ ) = u(Z,Zh) •
• As a particular case, take £ = £g, where g is another Hamiltonian. Then £gh = u)(tg,£,h) = -uj(£h,£g)
=
-ih9-
2.1.14. Definition. The Poisson bracket of two Hamiltonians is {h,9} = th9 = u{£h,£g) • 2.1.15. Proposition. £,{g,h} - [£s,£/>]-
Hamiltonian
27
Structures
Proof. The form u; is closed. Therefore,
V£ o = (dw)(£9,&,0 = £ M a , 0 - & w f o , 0 + M£ 9 >&)-w(fe.&]>0+ "(&»£]>&) -u([Zg,Zh],0 + [Zg,Z]h-[Zh,Z]g = -ttgh + tZh9 + Z{g,h}-u>([Zg,ZhU) = -£{ M - i(fo, &])w)(0 • Since £ is arbitrary, this yields d{g,h} = -i([£ 9 , &]) which is equivalent to the required identity.
•
2.1.16. Proposition. The Poisson bracket has the following properties: (i) {g,h} = (ii) {f,g-h}
-{h,g} = g{f,h}
(iii) {h1,{h2,h3}}
+
h{f,g}
+ c.p. = 0,
where c.p. symbolizes adding of all the cyclic permutations. Proof. Only (iii) (the Jacobi identity) needs verification: 0 = (dw)(&i>&2>&3)
= CfciW(Ch2,^3) - w ( [ 6 l l , £ / l 2 ] , 6 i 3 ) + c p . = Chi {^2, h3} - [£hl, ^h2]/l3 + C.p. = {/ll, {/l2, h3}} - £{huh2}h3
+ CP-
= {hi, {h2, h3}} - {{hi,h2},
h3} + c.p. = 2{hi, {h2, h3}} + c.p.
• The Poisson bracket turns the space of all functions into a Lie algebra. Proposition 2.1.15 means that the mapping h H-> £h is a homomorphism of the Lie algebra of functions with the Poisson bracket as a commutator to the Lie algebra of vector fields.
28
Soliton Equations
and Hamiltonian
Systems
2.1.17. Proposition. If / is a function, then, by virtue of the differential equation dtx = £H(X),
dtf = {n,f} holds. Proof. The differentiation with respect to the parameter t by virtue of the equation is equivalent to the action of the vector field fa:
dtf = faf = {H,f}.
• 2.1.18. Corollary. A function / is a first integral of the equation if and only if it commutes with the Hamiltonian, i.e. {%, / } = 0. 2.1.19. Definition. Two functions are in involution (i.e. are involutive) if {/,(Ha,Hp) =
(Ha,/3).
A question arises: what are the conditions on H equivalent to the fact that w is symplectic? H must be skew symmetric. The condition that w is closed is more complicated (see [GDor79, 80, 81] for more detail). 2.2.2. Definition. The Schouten bracket of two skew symmetric mappings H, J : T*M -» TM is a trilinear mapping [H, K] : T*M x T*M x T*M -> T{M) {!F{M) is the ring of functions on M) defined by Vai,a2,a3
eT*M
[H,K]{a1,a2,a3)
=
{KLHaia2,a3) + {HLKaia2,a3)
+ c.p.
Hamiltonian
29
Structures
2.2.3. Proposition. The form w is closed if and only if [H, H] = 0. Proof. Let £j = Hati, i = 1,2,3.
dw(6.6.&)=6w(6,6)-w([6.6],fo) + cp. = Hai{Ha2,a3)
- {[Hai,Ha2],a3)
+ c.p.
Transform the first term using 2.1.6: d w ( 6 , 6 , £ 3 ) = ([Hai,Ha2],a3)
+
- ([Hai,Ha2},a3) = (HLHaia3,a2) It remains to note that ai,a2
(Ha2,LHaia3) + c.p. = -(LHaia3,Ha2)
+ c.p.
+ c.p. = - [ H , # ] ( a i , a 2 , a 3 ) . and 0:3 are arbitrary.
•
2.2.4. Proposition. The correspondence / i-> ^/ can be expressed as
Proof, d/ = — i(£/)w is equivalent to £/ = Hdf.
•
2.2.5. Proposition. {f,g} =
{Hdf,dg).
Proof.
D 2.3
Variational Principles
2.3.1. What are usual sources of symplectic forms? For example, the natural symplectic form in the cotangent bundle is well known. We shall not speak in detail about this form since we do not use it directly. Briefly the main point is the following. Elements of the cotangent bundle are pairs: a point of the manifold, x 6 M, and a covector, i.e. an element a € T*M at the same point. The symplectic form is defined on the tangent space to the cotangent bundle at its point (x, a). The tangent vector will be given if we specify a shift of the point x, i.e. an element £ G TXM, and a shift of the covector a. A shift of the covector is a covector itself. Thus, elements of
30
Soliton Equations and Hamiltonian
Systems
the tangent space to the cotangent bundle at a point (x, a) are pairs (£, (3). The natural one-form on this space is LJ1 = (£, a). Then the symplectic form u> will be ui = dui1. 2.3.2. Now we describe another source of symplectic forms: variational principles of mechanics. We look for an extremal of the functional of action S = J Adt where the Lagrangian A depends on the coordinates {a:,}, i = 1 , . . . , n on the manifold as well as on their derivatives with respect to the parameter t : {xi,±i,Xi,... ,x\ }. It is well known that this problem leads to the neccessary condition for the extremum: the Euler-Lagrange equation. Integrating by parts we obtain
0 = SS=
L
5Adt=
y
——SxPdt
J* ^ dx^
-fe^-**- /;E (SA } => <j- = 0> ,
T—Sxidt OXi
SA n « = l , . . . , n ; 0 — = 0,
x =
{xi,...,xn).
Moreover, as we are going to show now, this procedure also yields the symplectic form and the Hamilton representation of the equation SA/ Sx = 0. Not making this notion more exact, we assume that the Lagrangian is nondegenerate, which is a generic case. For example, we assume that the highest derivatives do not enter the Lagrangian linearly and the order of the Lagrangian cannot be reduced by integration by parts. In the nondegenerate case the variational system is of the CauchyKovalevsky type, for which the highest derivatives can be expressed in terms of the lower ones. For example, when n = 2, 0 = 5A/6x1=ax^+bx^+N2)
+ .-.
0 = 6A/6x2 = cx[N>+N2)+dx?N2)
+ ---,
where the coefficients a, b, c and d as well as of the remaining terms contain derivatives of lower orders. Let JVi > N2 and differentiate the second
Hamiltonian
31
Structures
equation Ni — N2 times: B (*+^+i> + d4
2JVa+1)
+.
cx^+dx^+N2)+....
0=
If I"bdI ^ 0 (the condition must be included into the notion of nondegeneracy) then the first and the last of all the equations written here permit to express x\ 1' and x2 1 in terms of the lower derivatives; the rest of the equations give expressions for x2 x 2~ ',..., x2 in terms of the 1_ 2 lower derivatives, until only x\,... ,x\ ,x2, • • • ,x2 ~ a r e left. Thus, the variables x*f\
i = l , . . . , n ; j = 0,l,...,2Ni-l
(2.3.3)
can be accepted as the coordinates in the phase space of the equation 5A/6x = 0. A vector field £ corresponds to the equation 5A/Sx = 0 in the phase space and its action on the coordinates is the following: £x\J' = x\3+ ' , i.e. this is the differentiation dt. However the "extra" derivatives x\ '' must be eliminated with the help of the equations 5A/5x = 0. This implies that the action of £ on an arbitrary function f(xf) is the differentiation dt with the elimination of the "extra" derivatives. Let us consider differential forms Q, = Y',a^\'dx) A dx\' A • • • in the phase space. The Lie derivative of this form in the direction of a vector field £ is L&
=£(^:::)^fc)
A
• • • + E < - : : ^ f c ) ) A ••• + •• •
with the elimination of the extra derivatives. 2.3.4. Return to the procedure of integration by parts in the process of deducing the variational equation. This integration means that M is represented in the form 8A = ^2 AiSxi + dtu).
(2.3.5)
32
Soliton Equations and Hamiltonian
Systems
The first term contains only variations of the coordinates {XJ} and not of their derivatives; the second term is a derivative dt of a form a/ 1 ) = 53 o^Sx\ (pt acts on both the coefficients a] and the differentials dt5x\ ' = 5xf+1)). The form bjt1' was insignificant when we deduced the equation 5A/5x = 0; now it plays the decisive role. Equation (2.3.5) is an identity. If the extra derivatives are eliminated with the help of the equation 5A/Sx = 0, then dt turns to the Lie derivative Z/{ and Eq. (2.3.5) takes the form (we write here the more common symbol d instead of 5): dh = L ^
l
\
w™ =Y/a\1)dx(ij),
j < 2Ni.
(2.3.6)
Put tj = dw(-1\ This is a closed form which is nondegenerate in the generic case, i.e. symplectic. 2.3.7. Proposition. The equation 5A/Sx = 0 can be written as a Hamilton equation dW = —i(£)u> with respect to the form u introduced above. The Hamiltonian is ft = - A + i(£)w (1) (i(£)w (1) being calculated thus: iiQdx^ nation of the extra derivatives).
(2.3.8)
= £x\3) = x\3+1)
with the elimi-
Proof. Let us rewrite Eq. (2.3.6) as
dh = (di(0 + t(O = cL/ 1 ' we
2.3.11. Proposition. The symplectic form w can be written in the "coordinate-momentum" variables as
u, = 5 > « A ^ . 2.3.12. Corollary. The variational equation SA/6x = 0 in the same variables has the form
x? = dH/dp? P(P = -dUjdxf,
i = 1 , . . . ,n; j = 0 , . . . ,Nt - 1,
with the Hamiltonian "H = — A + ^2P\ xi expressed in terms of the canonical ones.
2.4
where the old variables are
Symplectic Form on an Orbit of the Coadjoint Representation of a Lie Group
2.4.1. Another example of a symplectic manifold yields an orbit of the coadjoint representation of a Lie group (see also [Arn74], App. 2). Let G be a Lie group and Q its Lie algebra which can be identified with TeG. There is a homomorphism of the group G to the group Aut Q of linear nondegenerate transformations of the linear space Q which is denoted as
34
Soliton Equations and Hamiltonian
Systems
g € G H-» Ad{g) € Aut Q. We have Ad(gh) = Ad(g) Ad(h). The mapping g >-t Ad(g) is the adjoint representation of a group. a Every representation of a group generates a representation of its algebra. Thus we obtain the adjoint representation of a Lie algebra, a € Q >-» ad(a) € End Q. We havead([a,/3]) = [ad(a),ad(/3)]. The adjoint representation of a Lie algebra is expressed by a simple formula V a, (3 £ Q, ad(a)/3 =
[a,0\. Let Q* be the dual to the linear space G, i.e. a coupling a e Q, m e Q*; (a,m) € R ( C ) is defined. This coupling is a bilinear function of a and m. The adjoint operators Ad*(g): Q* -» Q* and ad*(a): £* -» Q* are defined as usual: (m,Ad(g)a)
= (Ad*(g)m,a),
(m,ad(/?)a) = (ad*(/3)m,a).
Obviously, Ad*(ff/i) = Ad*(ft) AcTOO, ad*([a,/3]) = [ad*(/J), ad*(a)].
(2.4.2)
The space Q* is called a Lie coalgebra, mappings g H-> Ad*(g), and a i-» ad* (a) are coadjoint representations of a group and its algebra, respectively. 2.4.3. Let mo €
