Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1756
3 Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Peter E. Zhidkov
Korteweg-de Vries and Nonlinear Schrödinger Equations: Qualitative Theory
123
Author Peter E. Zhidkov Bogoliubov Laboratory of Theoretical Physics Joint Institute for Nuclear Research 141980 Dubna, Russia E-mail:
[email protected] Cataloging-in-Publication Data applied for
Mathematics Subject Classification (2000): 34B16, 34B40, 35D05, 35J65, 35Q53, 35Q55, 35P30, 37A05, 37K45 ISSN 0075-8434 ISBN 3-540-41833-4 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2001 Printed in Germany Typesetting: Camera-ready TEX output by the authors SPIN: 10759936 41/3142-543210 - Printed on acid-free paper
Contents
Page
Introduction
I
Notation
5
Chapter
1.
Evolutionary equations. Results
1.1 The
on
existence
9
(generalized) Korteweg-de
Vries equation (KdVE) Schr6dinger equation (NLSE) blowing up of solutions
10
1.2 The nonlinear
26
1.3 On the
36
1.4 Additional remarks.
Chapter
2.
37
39
Stationary problems
2.1 Existence of solutions. An ODE
42
approach
2.2 Existence of solutions. A variational method
49
2.3 The concentration- compactness method of P.L. Lions 2.4 On basis properties of systems of solutions
56
2.5 Additional remarks
76
Chapter 3.1 3.2 3.3
3.
Stability
Stability of Stability of Stability of
of solutions
79
soliton-like solutions
80
kinks for the KdVE solutions of the NLSE
90
nonvanishing
as
jxj
3.4 Additional remarks
Chapter
4. Invariant
62
94 103
105
measures
4.1 On Gaussian
measures
4.2 An invariant
measure
in Hilbert spaces
for the NLSE
4.3 An infinite series of invariant 4.4 Additional remarks
oo
measures
107 118
for the KdVE
124
135
Bibliography
137
Index
147
Introduction During differential
large
the last 30 years the
theory
equations (PDEs) possessing
of solitons
the
-
solutions of
special
a
partial
of nonlinear
theory
kind
field that attracts the attention of both mathematicians and
-
has grown into
physicists
a
in view
important applications and of the novelty of the problems. Physical problems leading to the equations under consideration are observed, for example, in the mono-
of its
graph by V.G. Makhankov [60]. One of the related mathematical discoveries is the possibility of studying certain nonlinear equations from this field by methods that these equations were developed to analyze the quantum inverse scattering problem; this subject, are called solvable by the method of the inverse scattering problem (on see, for
example [89,94]).
PDEs solvable
At the
by this method
is
time, the class of currently
same
sufficiently
narrow
and,
on
known nonlinear
the other
hand,
there is
The latter
of differential
called the qualitative theory equations. of various probthe includes on well-posedness investigations particular approach such solutions of as the behavior stability or blowing-up, lems for these equations,
approach,
another
in
dynamical systems generated by these equations, etc., and this approach possible to investigate an essentially wider class of problems (maybe in a of
properties makes it more
general study). In the
present book, the author
qualitative theory
are
on
about twenty years.
So, the selection of the material
the existence of solutions for initial-value
travelling problems
or
standing waves)
of the
stability
of
substituted in the
are
solitary
are
four main
problems
topics.
for these
equations,
special (for example, equations under consideration,
waves, and the construction of invariant
dynamical systems generated by
the
Korteweg-de
is
These
kinds
when solutions of
problems arising
studies of stationary
for
during
related to the author's scientific interests. There
results
and methods of the
equations under consideration, both stationary and evolutionary,
of
that he has dealt with
mainly
problems
some
surveys
Vries and nonlinear
measures
Schr6dinger
equations. We consider the
following (generalized) Korteweg-de +
Ut
and the nonlinear
f (U)U.,
+ UXXX
equation (KdVE)
0
Schr6dinger equation (NLSE) iut + Au + f (Jul')u
where i is the
imaginary unit,
and
in the
complex
=
Vries
second),
u
u(x, t)
=
t E
R,
x
is
an
=
0,
unknown function
E R in the
case
(real in the first
of the KdVE and
x
E
case
R' for N
the NLSE with
a
positive integer N, f (-)
is
a
smooth real function and A
=
E k=1
P.E. Zhidkov: LNM 1756, pp. 1 - 4, 2001 © Springer-Verlag Berlin Heidelberg 2001
82 aX2
k
2
Laplacian. Typical examples, important for physics, of the functions f (s)
is the
As 2) respectively,
the
are
and
following:
2 as
Isl"
value
initial-boundary for u
travelling
e `O(w, x)
=
the
waves u
in the
equation (it
what
being
be called the
solitary
(as JxJ
for the
NLSE,
-+
oo
dealt
we
wt)
in the
NLSE,
0
is
supplement
with
_
Loo
some
+
A similar
of existence and
0(k)(00)
=
(k
0
=
nontrivial solutions
integer
any
argument
r
occurs
Let
kinds).
In this case, the
us
typical
Ix I,
has
can
the
0,
possessing limits
Ej
X
=
00
as x --+
into the
waves
of the second order:
le,
0.
be
solitary
waves
solution of
I roots
on
satisfying solved
(see Chapter 2).
(for example, on
f interesting for our
r
>
for functions
the argument
us
problem which,
the half-line
proving
of
generally speaking, non-uniqueness
depending only
result for functions
1,
==
conditions of the
sufficiently easily
when such solutions exist
exactly
the method of the
of
consider solutions
a
into
function,
if necessary,
expression for standing
=
In this case,
We consider two methods of are
real
a
waves
notation, specifying, follows, the solutions of these kinds will
f(1012)0
uniqueness
I > 0 there exists =
standing
boundary conditions, for example,
0, 1, 2)
Difficulties arise when N > 2.
the above
is
0
Chapter
expression
arises for the KdVE. For the KdVE and the NLSE with N
problem
problem
type
substitute the
c R and
w
elliptic equation
061--
the
we
bounded function
a
nonlinear
following
Cauchy problem and
this
just
In what
with).
if
of the
of the KdVE and
case
where
NLSE). Substituting
A0 which
-
of the
waves
obtain the
we
It arises when
problem.
O(w, x
case
positive constants).
v are
well-posedness
is convenient to introduce
is
equation
=
the
on
and
a
for the KdVE and the NLSE used further. In
problems
the stationary
qonsider
we
(where
1 contains results
Chapter 2,
e-a.,2
+S21
1
is the as a
f
of
=
W.
following:
for
r
function of the
0.
the existence of
solitary
These
waves.
qualitative theory of ordinary differential equations (ODEs)
and the variational method.
As
an
example of
the
latter,
concentration- compactness method of P.L. Lions.
In
touch upon recent results
a
on
the property of
being
briefly
we
addition, basis
in this
consider the
chapter
(for example,
in
L2)
we
for
systems of eigenfunctions of nonlinear one-dimensional Sturm-Liouville-type problems in finite intervals similar to those indicated above.
Chapter Lyapunov set
3 is devoted to the
sense.
X, equipped
Omitting with
a
some
distance
stability of solitary waves, which is understood
details, this
R(., .),
means
there exists
that, a
if for
unique
an
arbitrary
solution
u(t),
uo
in the
from
t >
0,
a
of
3
to X for any fixed t >
equation under consideration, belonging
the
to X for any fixed
T(t), belonging R, if for
any
>
satisfying R(T (0), u(O)) < b, one Probably the historically first result on the stability
0.
that obtained
A.N.
by
for all
u(t), belonging to R(T (t), u(t)) < C for
has
Kolmogorov,
the one-dimensional
of
solitary
stability
a
solitary
case a
of
kink for
a
is called
wave
in
our
nonlinear diffusion
a
kink if
a
waves was
[48]:
I.G. Petrovskii and N.S. Piskunov
terminology, they proved (in particular) equation (in
solution
a
called stable with respect to the distance
X for any fixed t > 0 and
all t
0, then
0 there exists b > 0 such that for any solution
>
e
t > 0 is
0' (w, x) 0
0
X
x).
Let
introduce
us
functions of the
in the real Sobolev space H1
special distance
a
argument
of
consisting
by the following rule:
x,
p(u,v)=
JJu(-)-v(-+,r)JJHi.
inf ,ERN
If
we
for
call two functions
some 7-
E
and
u
from
v
H1, satisfying
set of classes of
R, equivalent, then the a
stability of solitary
waves
smooth
family
of
any two solutions t
0 in the
=
solitary
O(wl, x
sense
W2.
At the
same
distance p, then T.B.
Benjamin
stability of
they
in his
the
many authors and
For be
solitary
taken in the
and
wit)
same sense
time, can
if two
O(W2, X
-
the parameter
f (s)
form we
=
of
solitary
paper
first, because
usually
possesses
(a, b).
Therefore
E
have
close at t
=
velocities wi
0 in the
to be close for all t > 0 in the
has
proved
the
stability
of
solitary
wave.
Later, his
approach
point
Sobolev spaces,
or
non-equal
sense
solitary
was
of the
same sense.
with respect to the distance p. He called this
s
a
[7]
w
Lebesgue
as
they
waves are
with the
close to each other at the
L02t),
for all t > 0 if
easily verified
be
pioneering
the usual KdVE with
the
-
second,
on
of standard functional spaces such
cannot be close in the
and
depending
waves
the KdVE
r)
-
investigate the
of the KdVE with respect to this distance p;
the KdVE is invariant up to translations in x;
v(x
=_
equivalent functions
metric space. For several reasons, it is natural to
distance p becomes
a
u(x)
the condition
of
waves
stability
developed by
shall consider their results.
waves
of the
following
NLSE,
the distance p should be modified. It should
form:
d(u,v)=infllu(.)-e"yv(.--r)IIHI
(u,vEH')
T"Y
where H' is
only
now
the
complex
space,
-r
R' and
E
7 E R.
To
clarify
remark here that the usual one-dimensional cubic NLSE with
two-parameter family ob (x,
where
w
> 0
t)
and b
family, arbitrary
=
this
f (s)
=
fact, s
we
has
a
of solutions
V-2-w exp I i [bx
are
-
(b
real parameters.
close at t
=
0 in the
2 _
W)t]
Therefore,
sense
cosh[v/w-(x two
-
2bt)]
arbitrary
solutions from this
of the distance p, cannot be close for all
4
t > 0 in the any two
standing
close in the
of the
waves
close at t
NLSE,
parameter
of the distance p for all t
sense
above
family 40(x, t)
>
0 in the
=
0. At the
to each
cannot be
other,
time, the functions of
same
in the
stability
By analogy,
W.
-
of the distance p and
sense
nonequal
w,
the definition of
satisfy
V
to different values of
they correspond
to two values of the
corresponding the
if
same sense
of the distance
sense
d. In the two
cases
of the KdVE and the
condition for the
necessary")
stability
of
NLSE,
present
we
solitary
a
sufficient lim
satisfying
waves
(and O(x)
"almost 0 and
=
1XI-00
O(x) for
a
is called the
0, that
>
nonlinearity
of
general type
Next,
consider the
Confirming
this
the function
f
of
point
of
prove the
for
non-vanishing
of
our
Chapter 4,
JxJ
as
-+
oo.
We present
theory
physics.
waves.
of
a
equations.
If
For the
NLSE,
energy
and, for
we
construct
higher
that kinks
many our
always
are
stable.
assumptions
on
we
of
have
a
recurrence
an
present
case
an
a
new
constructing
attention
theorem
on
such
one
means
the
the
theory
application
con-
is well-known in
corresponding
measure
this
measures
in the
stability according
phenomenon
explains
measure
interesting
invariant
phenomenon which it
of the
waves
and
important applications
bounded invariant
invariant
solitary
remain open in this direction.
the Fermi-Pasta-Ularn
we
of
stability of
dynamical system generated by
the KdVE in the
scattering problem,
a
Roughly speaking,
By computer simulations,
system, then the Poincar6
with
problem
We concentrate
trajectories
many "soliton"
is satisfied.
> 0
of kinks under
stability
questions
It is the Fermi-Pasta-Ulam
of nonlinear
Poisson of all tion.
deal with the
we
dynamical systems.
the
many
equations. These objects have
nected with
stability
3 is devoted to the
Chapter
type. It should be said however that In
dw
of kinks for the KdVE with respect to the distance
widespread opinion
a we
d Q (0)
to the distance
general type.
The last part of
NLSE
view,
if the condition
NLSE)
stability
there is
Among physicists
p.
physical literature. Roughly speaking,
solitary wave is stable (with respect
a
p for the KdVE and to d for the we
in the
Q-criterion
was
for
to
equa-
observed for
our
dynamical
phenomenon partially.
associated with the conservation of
when it is solvable
by
the method of the inverse
infinite sequence of invariant
measures
associated
conservation laws.
The author wishes to thank all his
colleagues
and friends for the useful scien-
tific contacts and discussions with them that have contributed appearance of the
present book.
importantly
to the
Notation Unless stated the
otherwise,
of the KdVE and
case
the spaces of functions introduced
I for the KdVE and N is
(X1)
X
...
XN)
E
always real
in
for the NLSE.
complex
Everywhere C, C1, C2, C', C",... N
are
denote
a
positive
constants.
positive integer
for the NLSE.
RN.
N
E
A
i=1
R+
=
For
a
8'Xi
is the
[0, + oo). measurable domain Q C RN
of functions defined
Lp
=
Laplacian.
Lp(RN)
on
D with the
and
Mp
=
norm
for any g, h E
L2(9)-
00
00
12
ja
=
=
(ao,aj,...,a,,...)
:
an E
1: a2
R, 11al 1212
n
0
Oxiat-
aXklv N
is the set of
1, belonging
IXk8I+mU(XI0 I
I
...
< 00.
V
=
8
8
8XI
8- N
)
is the
gradient.
A > 0, Hpn,,(A) infinitely differentiable functions defined in For
of
(
equipped
an
integer
with the
n
> 0 and
is the
completion
R and
norm
I 2
A
I I U I I Hpn,,, (A)
-U( =10I -2(X) +
dnu(x)
of the linear space
periodic with the period A,
2
_) 1
dx
1
Chapter
1
Evolutionary equations. existence
on
In this
chapter
we
consider several results
lems for the KdVE and NLSE that remarks to this prove the
chapter,
we
on
the
[to, T], satisfy
segment
the
well-posedness of initial-value prob-
used in the next sections.
are
mention additional literature
result, generally well-known, which
Gronwell's lemma. Let a
Results
is
on
this
intensively exploited
inequality
y (t) :!
aIy (s)
ds +
b,
[to, T],
t E
to a
and b
are
positive
Then,
constants.
t
b
y(s)ds
b
a(t-to)
0 the
C((-T,T);H 2) n C'((-T, T); H-').
In
THE
1.1.
(GENERALIZED)
addition, if u(., t)
is
KORTEWEG-DE VRIES
H'-solution of the problem
a
(1. 1. 1), (1. 1. 2),
00
I 2 u'(x, t)
El(u(.,t))=
and
f 7(s)ds,
F(u)
0
t G
R,
i.
the
e.
junctionals E0
and
El
place we
we
initial data. Instead of it
with respect to the
shall
exploit this
solutions in the
case
2 and
Chapter
TI, T2
=
independent of
Cauchy problem certain result for the
for the standard KdVE with
x
problem
f (u)
=
u;
definition of
following
periodic.
are
Hpn,,(A) for
G
uo
u,
We call
0.
a
We introduce the
4.
when the initial data
f (u)
for the
consider
spatial variable
result in
Definition 1. 1.4 Let >
determined and
conservation laws.
are
with
periodic
are
0
A result similar to Theorem 1.1.3 takes
periodic
F(u(x, t)) dx,
U
f f (s)ds
=
-
x
-00
U
7(u)
quantities
I
and
00
where
then the
11
00
u'(x,t)dx
Eo (u (-, t))
EQUATION (KDVE)
function u(-, t)
A
some
0 and inte-
>
C((-Ti, T2); Hln,,,(A)) n C1 ((_ T1 T2 ). Hne-3 (A)) a solution of the problem (1.1.1),(1.1.2) periodic in x with P the period A > 0 (Or simply a periodic Hn-solution) if u(., 0) uo(.) in the space t 1. holds the G in and, for sense (-Ti, T2), equality (1. 1) any Hpn,,,(A) of the space -3 Hpn,,r (A) after the substitution of the function u in it. ger
n
,
7
>
a
E
r,
=
As onto
earlier,
it is correct to
The result
the
on
well-posedness
considered in this book is the
Theorem 1.1.5 Let any
integer
into
about
> 2
n
and
f (u)
uo E
sense
of the
global
for
a
periodic Hn-solution
solution
(defined for all t E R). problem (1.1.1),(1.1.2) in the periodic case
=
u
so
Hpn,,(A)
that
deal with the standard KdVE. Then
we
there exists
This solution
that
continuation of
a
a
following.
of the problem in the
speak
wider interval of time and about
a
a
for unique global periodic H'-solution
continuously depends
any T > 0 the map uo
i
)
C((-T, T); Hpn,,,,(A)) n C'(( T, T); Hpn,,,-3 (A)). -
u(-, t)
on
the initial data
is continuous
from
In addition there exists
a
Hpn,,(A) sequence
of quantities A
Eo (u)
A
I u'(x)dx,
Ej(u)
0
I
2
U2(X)
U
X
6
3(X) dx,
0
A
E.,, (u)
1 2 [U(n)12 X
+ CnU
[U(n-1)]2 X
qn(U7
...
(n-2))
)U X
dx,
n
=:
2,3,4,...,
0
where Cn
periodic
are
constants and qn
Hn-solution
u(.,t) of
are
polynomials,
the
problem (1.1.1),(1.1.2) (with f(u)
such that
for
any
integer =
n
u)
> 2
and
a
the quanti-
EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
CHAPTERL
12
ties
Our
do not
E,,(u(., t))
Eo(u(.,
conservation laws
proof
depend
for periodic
Wt
of the
f(W)W.'
+
+ WXXX +
1.1.6 Let
be
f(-)
get
(x, 0)
=
Wo
we
consider the
(1.1.4)
e>O,
(1.1.5)
(x)
the
infinitely differentiable function satisfying
E S and
following
statement which
f(-)
Then, for
(1.1.3).
be
E
E
to Coo ([0,
take the limit
we
1.1.7 Let
Proposition estimate
first,
xER, t>O,
0,
=
X
an
(1. 1. 3).
At the second step,
fact,
(4)
6W
Then, for any uo unique global solution which belongs
we
are
following:
Proposition
a
junctionals Eo,...' E,,
problem (1.1.1),(1.1.2):
W
estimate
the
e.
of Theorem 1.1.3 consists of several steps. At
following regularization
and prove the
t, i.
on
Hn-solutions.
an
any uo
c --4
is,
problem (1. 1-4), (0, 1] n); S) for an arbitrary n 1, 2,3,.... =
+0 in the problem
of course, of
an
(1. 1.4),(1.1.5).
independent
S there exists
a
In
interest.
infinitely differentiable function satisfying E
the
(1. 1. 5) has
the
unique solution u(.,t)
the E
00
U C-((-n, n); S) of the problem (1. 1. 1), (1. 1.2). n=1
At the third step,
Now
we
using Proposition 1.1.7,
turn to
proving Proposition
Lemma 1.1.8 The system
we
prove Theorem 1.1.3.
1.1.6. We
begin with
the
following:
of seminorms I
00
P1,0(u)
II (
=
2
2
dx1
)
dx
00
generates the topology in the
00
I
and
1
po,,(u)
x21u2(x)dx -00
1
I
=
0, 1, 2,...
space S.
Proof follows from the relations 00
2
PM
21
1(u)
x
U(X)
(dM )
00
2
dm dx
u(x)
dxm
,
00
Cl""
[X2,dmu(x) ] dxm
-00
min m;211
:5
dx-
E k=O
jI -.
2
x
2(21-k) u 2
(x)dx
+
d2m-kU) (dX2m-k
f
dx. 0
dx
0,
n
(1.1-6)
2,3,4,...,
=
U0(X)j
the Fourier
Using
R,
E
X
(1.1.7)
transform,
easily show
one can
that
00
U
Wn E
n
=
2,3,4,....
M=1
into account
Taking from
get
(1.1.3)
and
00
00
+
(194 ) Wn
dX
2
-00
-
-C0
I(
c
92 Wn 0XI
)2
dx-
-00
00
I (Wn
a4 +
Wn)
OX4
19Wn-1
f(Wn-1) -51- dx
0 be
constant
continuously differentiable function f (-) sup
from
we
I u 1,, F2 (C, p, f, R1, T)
=
u(=-HI: jjujjj:5R2
arbitrary
Lemma L 1. 9.
and let For
an
set:
sup
I f'(u) I
1U1<W
and
F3 (C, p, f, R1, T)
=
sup
If" (u) I
lul<W
(here W large R3
oo
0.
in view
Then,
of the embedding of H1
into
C).
there exists R4 > 0 such that
Take
for
any
an
6
arbitrary sufficiently E
(0, 1],
an
arbitrary
1.1.
KORTEWEG-DE VRIES
(GENERALIZED)
THE
EQUATION (KDVE)
infinitely differentiable function f(.), satisfying (1.1.3) with f, R1, T) :! , R3
and p and such that F2 (C, p, an
solution
arbitrary
w(x,t;c)
and F3 (C, p,
C([O,T];S) of
E
the above constants C
f, RI, T)
+0
as
n,
m
---
+oo and
by analogy
convergence of the sequence
fUn}n=1,2,3.... (1.1.25),
Due to the estimates
u(-, t) Indeed, let
us
take
E H
2
and
arbitrary
an
t E
for t
in the space
0. These estimates
0
Proof.
ftn}n=1,2,3....
Suppose this C
Un(-,t)
u(.,t)
as n -4 oo
statement is invalid and there exist
E
in >
Q-T,T]; H 2). 0 and
a
sequence
[-T, T]
such that
JjUn(* tn)
-
U(',tn)112
>
6-
ftnk}k=1,2,3.... be a subsequence of the sequence Itn}n=1,2,3... converging to some oo and one can easily prove as in to E [-T, T]. But U(', tnj u(-, to) in H 2 as k Lemma 1. 1. 14 that un,, (. t,,,) u (., to) as k oo in H2 So, we get a contradiction, Let
--+
-*
-+
--+
7
and Lemma 1.1.16 is
easily follows
It in the
proved.0
sense
existence of
uniqueness
from Lemma 1. 1. 16 that ut (. 7
of the space H-1 for any t E a
generalized
solution of the
of this solution. Let
problem (1.1.1),(1.1.2) defined have for t E
[0, T2)
and
U'
=
U1
ul
in -
00
1 d 2 dt
I '0
(., t)
an
and
f (u (-, t)) u', (., t) uxxx (., t) t) [-T, T] and, thus, we have proved the -
Let
problem U2
('7 t)
be two
interval of time
generalized
(-Tl, T2)
where
U2: 00
2 W
(X, t)dx
W(f(U1)U1x -00
-
f(U2)U2x)dx
prove the
us
solutions of the
T1, T2
> 0.
We
CHAPTERL
24
EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
00
00
I W[ff(Ul)
-
f(U2))U1x
+
f(U2)wx]dx
I
C(T2)
0.
solution fin (x,
converge in this space to
corresponding generalized for any T
that,
solutions of the
0,
>
JjUn(* t)-U(*it)jj2=0For each number
n
E C- Q-T,
of the
t)
T]; S)
(1.1.29)
let iin E S be such that for 0
problem (1. 1. 1),(1. 1.2)
we
have 1
Jjfin(*)t)-Un(')t)jj2
t' E
arbitrary t" for all
C([-t', t"]; H1).
(0, Tj)
and t" E
sufficiently large
n
Let
(-Ti, T2)
and
solutions, respectively.
(0, T2).
numbers
Hl-solutions of the
We want to prove that
and that
(-, t)
u(-, t) as (0, t'] and H'-solutions Un (' t) can be un
---
In view of the above arguments there exist tj E
such that for all
continued onto the segment
sufficiently large
[-tl, t2l
and
are
numbers
n
i
bounded in H'
uniformly
with respect
1.2.
SCHR6DINGER EQUATION (NLSE)
THE NONLINEAR
[-t1j t2l
for t E
[-tl7 t2l. Then,
to t E
(t
have
we
>
31
0):
t
Un(*it)12
JU(* t) I
'5
Cn
+
C4
11
8)
U
-
Un
(*) 5) 12ds,
0
where u,,
(-, t)
Cn1
+0
-+
u
-+
(-, t)
and C4
oo
as n as n
Further, by embedding
by analogy for
constant > 0 and
=
0, hence
t
lu,,(*, t) 12 corresponding solution u(-, t) of the problem
for any t, there exists C'
from the interval of the existence of the
(1.2.1),(1.2.2). Therefore,
this solution is
0 such that
global,
and Theorem 1.2.4 is
Remark 1.2-6 With the proof of Theorem 1.2.4
ficiently
smooth function
f
we
proved.0
in fact have shown for
the existence of H-solutions
(where
I
=
a
suf-
2,3,4,...)
of
CHAPTERL
32
equation (1.2.4) if Now
we
EVOLUTIONARY EQUATIONS. RESULTS ON EXISTENCE
Hi.
uo E
consider
a
result
the existence of solutions for the NLSE in classes of
on
functions
nonvanishing as jxj --+ oo. First of all, it should problem for the linear one-dimensional NLSE with f =-= 0 the space with
an
C(R). Indeed,
arbitrary to
argument
E R and satisfies
x
in any left
half-neighborhood
We shall show the
our
to
C(R)
t
point
of the
=
to
as
(this example
problem (1.2.1),(1.2.2)
1 is
a
Cauchy
ill-posed
in
47-tT::t7
e
function of the
is taken from
in the spaces
regularity
It is known that the operator e-itD acts in the spaces as
=
linear NLSE because this function is unbounded
of the
well-posedness
[0, to) belongs
when for initial data uo additional conditions of the
the
and N
this follows from the fact that the function
0 for each t E
>
be noted that the
are
[80]).
Xk (i.
e.
assumed).
L2 and H k, k
=
1, 2, 3,
following integral operator: 00
Gto
I K(x
=
-
yt)o(y)dy (t 54 0), Goo
=
0,
-00
where for t
:
K(x, t)
0
=
(47rit)72
quadrant
Rez
if t
12. 8 2
0, Imz
We also set
be written
can
> 0
as
exp( 0-2) (here 4t
the root
if t > 0 and in the fourth
g(U)
(4rzt)12
lies in the first
quadrant Rez
>
0, Imz
SCHR6DINGER EQUATION (NLSE)
arbitrary (in
0 is
what
follows,
we
prove that the
33
improper integrals
here
We have
converging).
1111
I
:5
i,,
[, +00
lim a
O(x
2V-t-s)
-
S=Ce2
+
j
2 Vs
,2
I
I
lim
+ 2
-+oo
isf
It-01(x
2v't-s)+O(x
2v't-s)
-
2s 2
Ids
0 such that if
O(X
e
O(x)
converges to
any
x
=
=
0,
uo E
x
E
R,
1,
t E
(1.2.14)
X3
E R and this
(1.2.15) function
is
a
solution
of equation
(1.2.7). Proof. Let first
show that it is
a
a
function
u(., t)
solution of the
prove that this function
Clearly, g(u(., t))
E
E
C (I;
X3) satisfy equation (1.2.7).
problem (1.2.14),(1.2.15).
For this
satisfies'equation (1.2.14). C(I; X3). Let us show that L[Gtuo]
=
aim,
0 for t
54
We shall
we
have to
0. We have
00
,92
02
_tGtUoj
I K(y, t)uo(x
=
_X2
aX2
+
y)dy
=
(
Gt
d2 UO dX2
00
where the Let
right-hand
us
side is continuous.
show the existence of
the definiteness that t
>
0
(the
-2-[Gtuo]. at
case
t < 0
By setting
can
z 4t
be considered
and
by analogy),
00
Gtuo
=
(47ri)-l'
uo
I
(x
+ 2 v"t_z) + uo (x
-
2
e
vft-z) dz.
N/"Z-
0
This relation
yields formally 00
-[Gtuo] at
=
(4ri)-12
f e' 'u'(x 0
0
+ 2 vft-z)
u'0 (x
-
VIt-
accepting
-
2
vltz).dz.
we
get
for
SCHR6DINGER EQUATION (NLSE)
THE NONLINEAR
1.2.
For
c
> 0
have
we
Ci-. U0(x + 2v t_z)
I
35
u'0 (x
-
2v'-t-z) dz
-
e
=
-ie
icu'(x 0
2 v'tc)
+
u'(x -0
-
,
-
2v1t_`c) +
-
Vt_
-\/t-
0 C
I
'(x+2v' t_z)+u0'(x-2-\/t_z)
uo
j _.
dz,
V"Z-
0
uo(x)
where
of this
0
--->
equality
jxj
as
--
tends to
and, consequently,
oo
is
an
improper integral which
containing
uniformly
zero as c --+ oo
(1.2.8)-(1.2.10)
interval. Since due to estimates
zero, for any t
> 0
x
with respect
side
bounded
right-hand
side
in t from any bounded interval not is determined and
58-t [Gtu0j
R the derivative
E
right-hand
to t from any
the second term in the
uniformly
converges
and
the first term in the
at
00
,t[Gtuo]
=
u" 0 (x +
i(4ri)
2vlt-z)
2
uo'(x VZ_ +
2vft-z)dz
-
0 00
i(47rit)-
e
a2
JEWL, U/1 (y)dy
-[Gtuo] X2
i
=
0
Co
so
that indeed
for t < 0
can
-12jGtuo] at at
[Gtuo]
be made
=
0 for t
iGtu"0
t=o
=
>
0. As
we
noted
earlier,
the
proof
of this relation
by analogy. a2
0, then obviously
If t
-2-
L[Gtuo]
[Gtuo]
a 2
=
u". 0
Further,
and therefore there exists lim t-0
iu" and 0
L[Gtuol
t=o
0. These
a
due to the above
[Gtuo]
at
=
iu". 0
Hence,
arguments
there exists
arguments also imply that
t
L
I ijGt_,(g(u(-,s)))ds I
=-g(u(xt)).
0
Thus,
first statement that any X3_SolUtion of equation
our
(1.2.14),(1.2.15) Now
we
is
problem
(1.2.14),(1.2.15)
satisfies
proved.
prove that any
equation (1.2.7).
satisfies the
(1.2.7)
X'-solution of the problem
In view of the above
it suffices to prove that the linear
arguments,
homogeneous problem Lu=O,
xER, tEI,
U(X, 0) has
only
the trivial solution
X'-solution of this
problem
u
=-=
=-:
0 from C (I;
0
X3)
.
Let
in the interval of time I. 0')
d
Tt
jI
ux
(x, t) I'dx
=
0
us
suppose that
Simple
u(x, t)
is
a
calculations show that
EVOLUTIONARY
CHAPTERI.
36
for all t E I.
in view of the
Now
the function
Therefore,
equation, u(x, t)
call
the
function of solution
a
I
k
1,
=
on
is the
oo
0, depending only
such that this solution
> 0
(resp. T2*
==
k
2, g(u) be
=
and I be
u
the
complex-valued
generalized
contin-
zero.
solution
Illuoll1k,
on
(or
a
We
X'-
nonvanishing
this
1, where k(x) is
p >
(1, N 2)
It is known that if p E
> 0 in Q.
different solutions.
starshaped, then,
0,
=
bounded domain with
problem (11.0.4),(11.0.5)
By analogy,
a
0,
=
with
(11.0.2)
Olau
pairwise
RN, 0
Along following similar problem: AO
the
E
suppose that solutions
the behavior of these solutions N
x
f(X, 02)0
O(X) 11. The
0,
real-valued function. We also consider
AO and
f(02)0
+
2
in Q and
7Nq j2, N-2
w
problem
(see Example
has
an
and
w
0,
a
C'then
infinite sequence of
0 and the domain fl is no
nontrivial solutions.
11.0.1 and Section 2 of this
STATIONARY PROBLEMS
CHAPTER 2.
40
chapter),
only if
has nontrivial solutions if and critical in the literature.
fying the
N >
problem (11.0.1),(11.0.3) with
the
condition
f(02)
Jim
OP',
=
is called
=
N-2
f (02)
lim
> 0
satis0 and
--
1951-oo JOIN-2
f(02)
lim
if
W
nonlinearity f(o 2) 0 (or f(x, 0 2) 0)
k61-00
supercritical
1, and N+'
N-2
the
+oo is called subcritical if
=
p >
N+'. The exponent p*
p
3,
so-
among
publications
de-
>
w
O(x)
one can
+
be
a
we
1, are sufficient. To get (11.0.6) one may, first, multiply
=
i
sum over
=
over
RN
and, second,
1, 2,..., N and integrate the result
integration by parts.
identity
in
only the trivial solution O(x)
change
xi
case
into account
equation (11.0.1) by 0 with the further integration of the result
multiply the
our
the function
This is
particular yields =-
that the
0 if the function
valid, for example,
for
problem (H.0-1),(H.0-3)
NG(0) w
>
+
(N
0 and
-
2)0g(O)
f(02)
=
has
does not
101P-1
with
CHAPTER 2.
42
Existence of solutions. An ODE
2.1 At
STATIONARY PROBLEMS
first,
we
consider
substitution
u(x, t)
solitary
O(x
=
-
for the KdVE and for the NLSE with N
waves
ct),
-Wol Assuming we come
that
0(oo)
to the
f M01
+
a, where a+ and
=
1. The
=
into the KdVE leads to the equation
R,
E
c
approach
+
01"
a-
0.
=
constants, and that
are
0"(00)
0,
=
following equation: -wO
7(0)
+
0"
+
=
(II.1.1)
a
0
with
a
=
7(0)
and
-wa-
f f (s)ds.
It is clear that the substitution of the
general
a-
representation for solitary equation (II.1.1)
qualitative analysis First of
all,
waves
be solved
can
into the NLSE leads to
by quadratures.
f
note that if
solution of this equation is bounded be continued onto
of
a
equation (11.1.1)
derivative
0"(-)
is
is
equation. Of course,
it is
However,
simpler
half-interval
on a
of the
[a, b),
then it follows from this
bounded,
where
b.
point
Setting Oo
=
0(a)
+
if
0'(.)
Oo, 0'(b) similar
=
the
00,
Cauchy problem
reasoning
for
immediately get
we
a
several times not
then it
a
can
solution
0
the second
b
f 0'(x)dx
and
00
=
0'(a)
+
f 0"(x)dx
and
a
equation (II.1.1)
our
a
and
of this solution
a
considering
b,
0 for some b > a FI(o) < 0 if 0 E (a, b). Let us prove that if these assumptions are valid, then equation (IT.1.1) has a solution satisfying the conditions 0(oo) a. Indeed, we take =
=
=
and
=
an
arbitrary point
Xo E
R and the
following
O(xo) Then, by (1.1-1) 0"(xo) x0.
There cannot exist
function E
=
1
2 [01(X)]2
< a
+
=
b,
0, thus 0'(x)
a such that Fl(al) (resp. 0'(x) > 0) in all points where this solution exists and there is no finite limit lim O(x)). Indeed, the first claim follows from (11.1.2) and the lim O(x) (resp. as
x
continuable
--*
on
a
>
x
x
a,
=
it should
X__00
be
Fi(al) and F, (0)
as
=
0 that is
< 0
for
above, 0'(x)
0
a
point
a, >
satisfying Fl(al)
a
(a, a,) exists, then two cases are possible: if f, (a,) (resp. 0'(x) > 0) for all x E R and lim O(x)
E
0
0, then
get
we
0
0, then,
a,
(resp.
00
X
lim
=
=
=
O(x) tending
the above solution
to
a
+00
jxj
--+
oo
and
always f, (a,)
>
possessing precisely
equation (11.1.1) types.
can
These
of extremum.
point
of maximum
(since fi(al)
=
Fl'(al),
0).
An observation
two
one
important
for
us
following
have bounded solutions
are
from the above arguments is that
possessing
limits
as
monotone solutions and solutions with
We shall call the
corresponding solitary
waves
x
-+
00 of
precisely
one
only
point
kinks and soliton-like
solutions, respectively. Let
(11.1.1) 0 is x
E
us
prove that the ab ove- constructed solution
and the conditions
continuously R
(in
O(w,oo)
differentiable
as
a
=
a,
0(w,x)
>
a
O(w, x) satisfying equation x E R and 0'(W,xo)
for
function of the argument
view of the invariance of the
equation with respect
X
W
for
an
arbitrary
to translations in
CHAPTER 2.
44
x,
cannot state the
one
solutions
depending
b introduced above is
8)
6, wo + b is
wob
8
some
respect point
to
is
w
w).
0. But this follows from the
algebraic equation F, (w, b) the continuous
: 0, and, thus,
a
-
of
W
arbitrary family
=:
implicit
a
function of
of the solution
a
wo b >
-
0, f (0)
=
0 and
wo < 0
-
F, (0)
prove that
and there exist b
0 for
0
(0, b) Then, using -
the above
-
O(W, x)
-
with
fixed
a
TW
P
E
here
I 0'(w,x)dx
1'(0)
W
-00
If
f (0)
oo
are
and
0"
=
-00
>
v
well known.
0, then soliton-like solutions of the KdVE vanishing
They
form the
[ V\//-)-(( (_2( _+1 jv
-
where
2
sech(z)
C.+e-.
,
A > 0 is
a
as
following one-parameter family 2A 2t
Av
A sech
one can
00
f 0(wo,x)0'(wo,x)dx
2
0, f, (b)
=
methods,
H1 and
0"
d
(
2))]
_
_
_v-! )_ )(v
1)(v
+
+
2A2
real parameter and
w
(v+l)(v+2)
One
can
easily
this formula.
verify
The above considerations show that the one-dimensional equation vant to the =
7(b)
=
denote the above function with
we
(Wo
E
W
x0.
easily
N
of
function theorem because
0, where -2-F, (wo, r) I r=b i9r
differentiability
proved. Everywhere by 0'
as
W
Let
x
an
It is sufficient to prove that the parameter
locally continuously differentiable
>
solution of the
a
f (b)
for
with respect to
differentiability
the parameter
on
STATIONARY PROBLEMS
I is
problem
simple
NLSE with N
in
of the existence of
a sense.
> 2. In
takes the
we
begin
this and the next
problem (11-0.1),(11-0-3).
(11.0.1)
Now,
solitary to
study the
sections,
After the substitution
following
we
O(x)
(II.1.1)
rele-
for the KdVE and NLSE with
waves
multi- dimensional
case
of the
consider radial solutions of the
y(r),
=
where
r
=
jxj, equation
form: N- I
yll+
y'= g(y),
>
r
0,
(11.1.4)
r
where
g(y)
equation
First
we
(0, a)
f (y2)y
0y
obtain
hood of the
functions
=
and the
prime
y(0)
yo E
means
the derivative in
r.
We
supply
this
with initial data
a
point
result r
=
on
=
the local
a
y'(0)
< +oo.
=
well-posedness
0. The class of solutions
continuously differentiable
where 0
0 is
(z, Tkm+,)
depend
C(T;'k'+,)
>
independent
(z, zi)
E
r
where zi E
G(")
dr
is such that
but
on m
we
12
Y12 (r)
!
2(N
y
I
-
of
has 0
one
2
m.
We have
G(ym(T-k+,))
C,
0;
I ym(r) I : ' Co for some C, ym (.1) E (a,, a2) and G(y- (zi))
simplicity
of
C2(Tmk+,)-l
r z
if
r
for
d2 -). Thus, (II.I.11) holds. Finally, by (11.1.4) we have dr2 (rm, k ym(r) :5 -C3 < all r E (r', Tlk+,) such that ym (r) > d. In view of this fact and k (11. 1. 11) we get
G
rm k
and Lemma 11.1.3 is
Summing
the
-
Tmk+1
C,
a
k
points of
(11.1.9)
extremum lie from the outside
constant C > 0 such that
=
1, 2,..., 1,
m
and since the
=
1, 2,3,....
EXISTENCE OF SOLUTIONS. A VARIATIONAL METHOD
12.2.
Thus, the
function V has
continuous if
y(ro)
==
of solutions
dependence
0
imply
less than I roots. At the
no
same
49
time, the theorem
the parameter yo and the fact that
on
the
on
y(ro) =A
in view of the definition of -go that the function V cannot have
0
more
than I roots. Let
prove that
us
-9(r)
lim
0.
=
Suppose
this is not the
either
Then,
case.
r-00
the solution V is monotone for Zn can
be
only
negative
sufficiently large
In the first case, the
+00-
-4
for
=
y
or
a-,
graph
values of the argument
sufficiently large
number of roots of the solution V is y E
large
equal
in this
(a-,, 0) U(O, a,). Thus,
values of r, too. But the
yo, that for
sufficiently large values m solutions ythat lim
y(r)
=
asymptote which
an
of the solution
'g(r)
is
In the second case, since the
E(r)
is
negative
implies, by
problem (11.1.4),(11.1.5)
dependence implies
P(r)
r.
of the energy
negativeness
of solutions of the
continuous
contradiction
sequence of extrema
a
it should have extrema in the domain
1,
to
the energy
case
it has
or
the energy
Hence,
a,.
=
y
r
of this function has
cannot have
0, and Theorem H.1.3
on
more
is
for
sufficiently
the theorem
on
the parameter
than I roots. This
completely proved.0
r-oo
Existence of solutions. A variational method
2.2
In this section
we
shall consider
an
existence of radial solutions of the to the
case
f(02)
=
jolp-1,
problem (11.0.1),(11.0.3). So,
p > 1.
AO
=
wO
consider the
we
-
101P-10, =
Our result
the existence is the
on
Theorem H.2.1 Let
>
Lo
problem (11.2.1),(11.2.2) has radial solution We note that
right-hand sense
ul a
=
ul(r),
Loo
-
N > 3 be
0,
where
r
=
jxj,
with
'(rV)lr=r(,,)
Jr
critical
point
be =
a
0.
vo E
E R
(11.2.1)
,
(11.2.2)
0.
integer, and
p E
precisely
1 roots
Then, (1, N+2). N-2 any 1
on
the
half-line
problem (11.2.1),(11.2.2)
a more
general
kind
g(o)
the
1,2,3,...,
=
for the
we
present below
a
are
used when
real Hilbert space with in
for functions
r
a
> 0.
with the
g(O)
in
a
Then, if then
the
functional
Y(h) Ih=r(vo)vo
j(v)
-0.
proving Theorem 11.2.1.
a
H, and S
continuously differentiable function
S,
N
place
tinuously differentiable real-valued functional > 0
attention
101P`0.
Theorem 11.2.2 Let H be
r(.)
the
proving
our
problem
positive radial solution and, for
equation of
Two results which
Let
x
We restrict
following.
similar result takes
side of the
similar to
a
of variational methods to
application
=
on
norm
=
fh
11
-
E H
S such that
J(r(v)v)
J be
11,
I IhI I
:
for
considered
any
on
a
=
v
con-
1}. E
S has
S a
50
CHAPTER 2.
Proof is clear.
By conditions of
STATIONARY PROBLEMS
the theorem
d
J'(r(vo)vo),vo
=
=
0.
0
consider
Further,
such that
-y(s) d
0
=
an
arbitrary continuously differentiable
E S for
s
E
(-1, 1), 7(0)
E L.
w
Thus,
=
following.
H,'
and the
with h almost
Sketth of the Proof.
r(vo)-j'(0)
+
[75,76].
product
2 < q
C
0 there exists
00
h (r)
lim
(s) ds
(11.2-3)
n-oo
and for any a, b is continuous
on
:
0
r
(+oo)
hence
=
Also,
0.
we
h(s) ds
I
Rj)
I
N-1 r
Ih.,,(r)lldr
:5 DN Cq-2rq-2 (R)jjh,,jjj2,r'
R
Since here for
ciently large
R
an
>
arbitrary given 0 and a] I
n
=
6 > 0 the
1, 2, 3,
...,
Remark H.2.5 Theorem H.2.4 is
right-hand
the sequence
particular
a
side is smaller than 6 for suffi-
I hnln=1,2,3....
Now,
we
turn to
proving Theorem
of
case
Sobolev spaces of functions with symmetries obtained
by
compact in Lq-E3
embedding
P.L. Lions in
We consider the
H.2.1.
is
theorems of
[56].
following
spaces of
Hr(a, b) of functions u(jxj) from Hr' satisfying the condition or jxj : b (here 0 < a < b) and the space H,(b) of functions jxj u(jxj) b. Let H,(0, b) from Hr' satisfying u (I x 0 as I x H,. H,(b) and Hr(+ 00) Clearly u E Hr' for any u E H,(a, b). functions: =
0
the space
as
sequence
in the
of the domain
point
JxJ
elliptic equation (11.2.5)
function. If b
0
Co'
E
v
0 for
over a
C000 satisfying the neighborhood of zero. Taking v
=
is
classical,
at any
small
i.
that it
e.
point
b if b
I is arbitrary if p o(ri) q, N-2 N+2) N-2 N-2ri N-21 p(N-2)-4
Then,
since
0
C-
we
=
have
where qo
E
.
-
=
*
-
=
=
-
-
,
C(BR(O)) large
r
>
if p E
(1, N42 ) .In
the two latter
cases
0
E
Wr2(BR(O))
with
1, hence, 0 is continuously differentiable in BR(O), therefore it is
solution of
our
equation.
arbitrary
a
classical
54
CHAPTER 2.
Consider the
case
also observe that q-,
4
(N
p E
2
,
N+).
One
N-2
STATIONARY PROBLEMS
easily verify
can
that q,
>
We
qo.
p(?P(qo)). By analogy, one can show that, if r2 q2 < : , then 0 Lq2PRM) with q2 (p(r2) > qj, if r3 0(q2) 2 P P 0 E L, (BR(O)) with q3 W(r3) > q2, and so on. So, we get either infinite increasing and fqn}n=0,1,2.... satisfying rn < L for all n or a number no sequences 1rnjn=1,2,3 2 "'
;2 ,
when these
case
6 > 0 and all
n
because
it must converge to
verified that this map has
simply
remaining
cases
the
by
embedding
a
no
theorems
1, therefore by embedding theorems
classical solution of
equation (11.2.5)
and
proved.
arbitrary positive integer
and let
6 for
-
bounded, hence, be
can
In the
=
qo
continuously differentiable, thus,
the
L. First of all, in the 2
infinite it cannot be that rn < : 2
0 and a1+1
1. Let 0 < a, < a2
0,... and U2 < O)U4 < 07 (solutions U2, U41 exist because if u is a positive solution of the problem (11.2-5),(11.2.6), then -u is a negative solution). We denote by u(jxj) the function which is equal to Uk(IXI) in the domain ak-1 < JXJ :!, ak, ao
=
+oo.
=
....
the above- constructed radial solutions of the
=
=
...
k
1, 2,
=
...'
1+ 1. Let R be the
Then, according
ak.
of aA;,
i.e.,
set of all these functions for all values of the
(11.2.8) J(u)
parameters 0
=
1, 2, 3,
=
2
T' ) Jjujj,'
p+1
a' < a' < a' < 0 1 2
Lemma 11.2.6 There exists n
(il
-
on
sequence for this functional J considered
values of
R1,
=
the functional J is bounded from below
minimizing ing
to
...
to
converging
a
...
=
u
parameters
E R and for all values
the set R. Let
a1
0
only
be obtained
n such that ak+1
-
n ak > -
> C > 0 prove that a' 1 -
C,
(n
by analogy.
It suffices to prove that
inf
J(U)
--+
+00
UEHO'(b)nm, u:oo as
b
that too.
-4
+0.
Suppose
it is not
right. Then,
there exists
a
sequence
bn
--+
+0 such
110nill < Q2 for some minimizers On (here n 1,2,3,...). Then JO.jp+j :5 C2) Let Vn(X) where anOn(bnjxj) an > 0 are chosen for the functions vn to satisfy =
=
EXISTENCE OF SOLUTIONS. A VARIATIONAL METHOD
2.2.
condition
1 1 VnJJ21
with
(11.2.8)
a
0 and b
=
1.
==
55
Then, using embedding theorems,
get
we
therefore C3 IVVnl2, 2
Cap n- 10njP+1 4 P 2-
for
0
C3, C4
some
0
>
H,' satisfying (11.2.8),
E
of
independent
Hence,
n.
since
earlier
as
! C5
10,,Ip+l
> 0
for all
have
we
2
C6b,,P,-'
0 < an
0
one
> I
has IA
-I-, N
+
0. To
0.
For P E
(1, 1
M N
+
consider the function
VA
2
u(c, X)
e
N
(27ror2) T I U (or, .) 122
We have
=A and 1
E (u (o,,
x)) =20r2
1 IVU(j, X) 12
dx-
(p+l)+N
2
07
RN
Hence, E(u(u,
E(u(u, -))
becomes
Remark 11.3.3 more
general
-oo
kind. For
as
-4
a
for
+0 when p
sufficiently large
P.L. Lions in
example, -Au +
he
c(x)
>
0 and
f (x, u)
is
[57,58]
c(x)u
f (x, u),
=
I I-I--
something
=
like
1 u (1, x) jP+1 dx.
the relative compactness of any
cr
1 +
'
N
the
x
and,
for p E
(1, 1
+
N
0.
>
problems
of the
essentially
problem E R
N
0,
k(x) ju IP-'u
Remark 11.3.4 As P.L. Lions noted in his
providing
>
considered
investigated
U
where
p + 1
RN
negative
Really,
1
with
publications,
minimizing
k(x) the
>
0.
principal relation
sequence up to translations
as
58
CHAPTER 2.
in Theorem 11.3.1 is our
I,\
I,, + I,\-,, for
0
c
any
there
for which
U2k(X +Yk)dx>A-c,
k=1,2,3,...
BR +(0)
yl,
(here BR(O) fx E RN: jxj (ii) (vanishing) =
0;
(iii) (dichotomy) bounded in [V and
there exist
(0, A)
G
and sequences
fUk1}k=1,2,3....
and
JU2k}k=1,2,3,...
satisfying the following: I
u""
a
(UA;
-
+
U2)1 k
2N --+0
I (Uk1)2dX
lim k-oo
k--+oo
as
q
-
a
I (Uk2)2dX
lim
=
k-00
RN
for 2 0.
RN
With the Proof of Lemma 11.3.5 we, from Part I of
[57].
actually, repeat
the
proof
We introduce the concentration functions of
Qn(0
=
f
SUP YERN
of Lemma 111. 1
measures
U2 (x)dx.
y+Bt (0)
Then, jQn(t)Jn=1,2,3.... functions
on
R+
and
is
a
sequence of nondecreasing,
liM
Qn(t)
=
A.
By
nonnegative, uniformly bounded
the classical
result,
there exist
a
subse-
t- +00
quence f Qnk } k=1,2,3.... and
that lim
Qn, (t) k-oo
=
a
Q (t) for
function
Q(t) nonnegative
any t > 0.
and
nondecreasing
on
R+ such
2.3THE CONCENTRATION-COMPACTNESS METHOD OF RLIONS
Let
a
lim
=
t
Q(t). Obviously
[0, A].
E
If
a
0, then the vanishing (ii) takes
=
+00
fQnk(t)}k=1,2,3,.-
for the sequence
place
a
59
If
a
A, then clearly the compactness (i)
=
occurs.
Let
be
briefly
us
arbitrary. Then,
prove these two claims.
let
First,
a
0 and let
=
c
0 and R > 0
>
have
we
Qn,(R)
=
I
sup yERN
u',, (x) dx
sufficiently large
obviously
there exist sequences
where Mk
are
+01 Rk
--+
Ck
such that
positive integers,
+oo
-->
Qn- (Rk)
as
>
k
--+
6k
a
and
oo
for all
M
=
A.
Then,
fMklk=1,2,3
>_
....
)
Take Yk
Mk-
such that
I
2, (x) dx
U
=
n
k
Q,,,, (Rk),
1, 2, 3,
=
...
yj,+BRk(0)
Then, taking
an
arbitrary
>
c
0,
we
I
have for all k such that
n_(x + Un
y,,,)dx
>
A
-
el,
mk-:
m
c.
BR, (0)
Now,
R > Rk
place k
--+
get this relation for all
to
so
Consider the
case a
in this
Clearly,
case.
for all k
>
p(x)
k,,,. Then,
0, W
=-
0( ) and .
-
We have to prove that the
there exist sequences Rk
=
a,
we
Jxj
=
A
set
=
m
R,
= ...
< 1
I Qn, (4Rk)
and
there exists and
=
= ...
Rk_+j
get required
be cut-off
0 for W,,
'Ek
Q(m)1:5 m-1
we
172,3,..., Rk_+1 Let
1,
sufficiently large
a
proved.
(0, A).
positive integer
any
Q(+oo)
a
-
IQ,,,,(m)
m
it suffices to take
...'
dichotomy (iii)
+oo and Ck >
01
=
a
-+
0
as
=
17
M7 Ck-+l
El
-
Q(4m)l = ...
'Ek,
Ckm+l
m-'
E (uo), but then E Hence, JU0122 lim E (un, ), hence, (uo) k-oo quence
=
=
k
IiM
IlUn,'(* +Yk)JII k-00 uo
strongly
in H1
=
as
Iluolli, k
---
-
k-oo
therefore the sequence
oo.
This
easily yields
fUn' (* +Yk)}k=1,2,3.... k
k
converges to
the statement of Theorem 11.3.1.0
CHAPTER 2.
62
On basis
2.4 In this
section,
we
eigenvalue problems
suitable spaces of functions we
consider the
if
+
_U
as
segment similar
to
(11.0.1),(11.0.3)
L2 containing "arbitrary
f(U2)U
by
the
systems of eigenfunctions of one-dimensional
cases
on a
nonlinear
following
consider recent results obtained
example)
an
author which state that in certain nonlinear
of systems of solutions
properties
briefly (with
STATIONARY PROBLEMS
can
functions".
be bases in
More
precisely,
eigenvalue problem: AU,
=
U(O)
=
x
E
U(1)
(0, 1), =
u
u(x),
=
(11.4.1)
(11.4.2)
0,
U2(X)dx
(11.4.3)
0
where a
all
again
spectral
parameter. If
continuously A the
quantities
and
eigenvalue
that it is not clear
correspond At real
a
differentiable
a
are
is
pair (A, u), on
a
where A E R and
satisfies the
[0, 1],
u
eigenvalue.
first,
we
introduce
scalar
a
is
a
are
of this
and the
product
then
problem.
to the coefficient
partially
is
function twice
problem (11.4.1)-(11.4.3),
definitions which
some
Hilbert space with
u(x)
=
corresponding eigenfunction u(x) priori that only one eigenfunction (up
an
function, and A
smooth
given sufficiently
the
to
separable
real, f
we
call
We note
1)
can
known. Let H be
corresponding
a
norm
1 2
Definition 11.4.1. A system
for
an
Ih,,}n=0,1,2....
h E H there exists
arbitrary
a
unique
C H is called
sequence
of
real
a
basis
of the
space H
if
coefficients janjn=0,1,2,...
"0
such that
E anhn
=
h in the
of the
sense
space H.
n=O
Definition 11.4.2. A system the
anhn
equality
=
0, where
Ihn}n=0,1,2.... an
are
real
E H is called
coefficients,
linearly independent if
takes
place
in H
only for
=0
0
=
ao
=
a,
=
= ...
an
=....
In accordance with the papers
Definition 11.4.3. We call
a
[5,6]
basis
we
introduce the
Ihn}n=0,1,2... of
following
the space H
two definitions.
a
Riesz basis
of
00
this space
if the
series
E anhn n=O
co
when
F_ a 2< n
n=O
00.
with real
coefficients
an converges in
H when and
only
ON BASIS PROPERTIES OF SYSTE,MS OF SOLUTIONS
2.4.
Definition H.4.4. Two systems called
close in H
quadratically
f enjn=0,1,2....
if E Jjhn
-
11
en
2
C H and
0 the problem (11-4. * (11-4. 3) has a pair (An Un);
con-
a
and let
7
sisting of an eigenvalue An and
a
function
n
Un possesses
up to the
coefficient
(b) 112(01 1)-
the system
Before need also with
a
proving
1.
1
corresponding eigenfunction roots in the interval
of the function
I I 11. -
this
result,
We call
conditions
p(x)
> 0
Un; in
we
are
a
shall prove the
proved
(0, 1),
addition, Ao
of eigenjunctions fUn}n=0,1,2....
theorem of I.M. Gelfand
a norm
following
precisely
in
is
a
2 for any x E B12 (X2) Continue this process. We get a sequence f B,. (X,,,)}n=1,2,3.... of balls such that Bn,, (x +,) C 0 as BIn (xn) and p(x) > n for any x E B, (xn); in addition, we can accept that rn oo. Then, there is a unique xo E n nB,,,(xn). By the construction p(xo) > n for C
for all
> I
there
E
x
exists X2 E
-
--+
---+
n>1
any
integer It
n
> 0.
This contradiction proves the lemma.0
easily follows from
Lemma 11.4.6 that
functional in X is continuous. we
Indeed,
let
an
xo E X.
admissible lower sernicontinuous
Then,
since p(x) :5
p(x xo) +p(xo), -
have
p(x) On the other
-
P(xo)
:5-
Xx
-
x0)
0 there exists 6
that
P(xo) for all
x:
jjx
-
xoll
0 such that
P(XO) Choose 6
PN(XO)
2
for
2 x
xo
-
xoI I
-
II
8
0
by
the convergence of the series
en n=O
hnI 12. Clearly, continuity.
coefficients a,
We have
only
are
linear functionals in H.
to prove that if
f hl}1=1,2,3....
C H
Let
and lim 1
any fixed case.
n
Then,
the
corresponding
we can
assume,
coefficient a'
passing
--+
0
as
I
n
to
a
subsequence
--+
us
oo.
also show their
I I hI I
=
0, then for
00
Suppose
this is not the
if necessary, that
an'
!
co >
0.
STATIONARY PROBLEMS
CHAPTER 2.
66
Then, g'=
(a,,)-'h'= E bkek
+ en
n
0 in H
--*
I
as
But then
oo.
--+
H
1
as
--+
and
oo
E bke-k
--+
in
g'
kon
kon
bA;ek for
clearly g'
real coefficients
some
hence
bk,
k96n
en
+
1: bk ek
in
0
=
H,
kon
i.
get
we
e.
a
contradiction. Thus indeed coefficients
are
an
continuous linear func-
tionals in H. 00
E an(en
Let F
-
and F
hn)
The operator U is linear and it is
Uf.
=
n=O
determined
BR(O)
If
=
in H.
everywhere E H
11f1l
:
2
M211fJ12
e
0. There exists
a
number N
>
0
large
so
F,
that
I I en hn
=
=
(
=
.
identities
from another and
one
the result
integrating
over
the
segment [0, Y],
we
get:
7
f Ul(X)U2(X) [f(U2(X))
0 >
_
1
f(U2(X)) 2
-
An (a) + A'] dx.
(11.4.9)
0
But since
by
inequality
is
our
suppositionUl(X)
positive,
The property Lemma 11.4.9 is
We
i.
we
e.
An(a)
get
a
>
U2(X)
(r(n + 1))2
!
for
x
E
(0, Y)
,
the
right-hand
side of this
contradiction.
follows from the comparison theorem.
Thus,
proved.0
the notation
An(a) for the value of the parameter A from Lemma u,,(a,An(a), x)= Un(a, x). By Lemma 11.4.9 these definitions are correct
keep
11.4.9. Let
1
and
An (a)
>
0 for any
a
0 and
>
integer
n
> 0. Let also an
f Un' (a, x) dx.
(a)
0
Lemma 11.4.10 For any integer continuous
on
the
Proof. Let
half-line
a
n
> 0
the
function An(a)
is
nondecreasing
and
> 0.
> 0-
We shall prove that
An(al) ! An(a2)- Suppose the contrary, ul (x) un (a,, x) and U2 (X) Un (a2 x). By the properties (a)-(c) from the proof of Lemma 11.4.9, each of the functions ul(x) and 11is the point of maximum of U2(X) increases on the segment [0, 2(n+1) ] and x 2(n+1) each of them. By (11.4.8)., we have u',(xi) > U2(X2) for any y > 0 for which there exist i.
e.
a,
> a2
that An (a,)
U2 I
(X)
for all I
-
taking
x
=
2(n+l)
get 2
U1(X)U2(X)[f(U 2(X)) 1
0
_
f(U2(X)) 2
n(aj)
-
+
An(a2)]dx,
0
which is is
obviously
proved Let
contradiction because the
a
that the function
is
0 such that
ao >
for each
> ao
a
the half-line
on
>
An(ao)
d(a)
+0 as a ao + 0; 1) d(a) 2) u, (a, x) > un (ao, x) for x E (0, d(a)); 3) Un (a, d(a)) Un (ao, d(a)) and -Adx un (a, d(a)) -+
second
it follows from
there exists
ao
a
positive. So,
it
0.
>
the contrary, i.
An(a)
An(ao).
be considered
case can
(11.4.6)
(11.4.8)
and
that
0 such that
-
:--:
< -- - u, dx
(ao, d(a)).
Un(a, x) < Un(ao, x) in a right half-neighborhood of the point x d(a) (because 0,xx(a, d(a)) < u",xjao, d(a))). Then, as above, it follows from equality (11.4.8) that Un(a, x) < Un(ao, x) for all a > ao sufficiently close to ao and for all x E (d(a), ; '-+ ). Using the identity similar to (11.4.9) with the integral over the 2(n+l) segment [d(a), 2(n+l) we get a contradiction. So, the function ' n(a) is continuous, Therefore
=
n
n
and Lemma 11.4.10 is
Lemma 11.4.11 a
liM
0,
>
an(a)
=
proved.0
an(a)
is
0 and
+0
a
a
strictly increasing
lim
an(a)
=
continuous
function
on
the
half-line
+oo.
a-+oo
of the function
an(a)
follows from the
Of
) n(a) (see Lemma 11.4.10) and from the continuous dependence of solutions of the problem (11.4.6),(11.4.7) on the parameters a 12[ A. Further, as it is proved earlier, un(a, x) 0 as a +0 uniformly in x G [0, 1] (see the proof of Lemma IIA. 10), therefore The
Proof.-
continuity
continuity
--->
--*
liM a
an(a)
=
0.
+0
Let
us
prove that
lim
an
(a)
=
+oo. First of
all,
we
observe that
for all
(0,
x
2(n+l) ). such that u"
(-=
(0,
2(n+l)
and also
Indeed,
if
we
suppose the
(a, xo)
>
0.
But
n, xx
f(U2)U
addition, by
-
our
An(a)u
is
a
n
contrary, then there exists
u',x (a, xo)
>
0
as
it
was
:!' 0
xo
E
indicated earlier
n
nondecreasing
supposition f (u 2(a, xo))
u",xx (a, x) n
a-+oo 1
-
function
on
the half-line
An (a)Un (a, xo)
>
0.
u
E
Hence,
[0, +oo); we
in
get that
ON BASIS PROPERTIES OF SYSTEMS OF SOLUTIONS
2.4.
Un',.,(a, x) all
(xo, 11,
E
x
i.
e.
have
we
contradiction.
a
So, un,,(a, x)
for
< 0
2(n'+I))'
E(01
x
0 for all
>
71
to prove that
Now,
lim
a,,(a)
=
+oo, it suffices
to show that
u,,(a,
+oo
a
1
2(n+l)
(because, as it is proved above, Un(a, x) is a concave function of x segment [0, n+1 T' ]). Suppose that for a sequence ak +oo the following takes place: Un(ak7 2(n+l) ; '+- ) :5 C < +oo. Consider separately the following two cases: A. +oo and B. f (+oo) < +oo. f (+oo) +oo
+oo
as a --+
the
on
-+
=
A. Let
f (+oo)
would get from
we
=
Then
+oo.
(11.4.8)
the functions Un (ak ,
Therefore,
An(a) u,(ak, k-oo
that
in the interval
+oo
-*
ing
Un(ak7 X)
for all
B. Let
--
oo
uniformly
arguments based
So,
the
f (+oo)
A is
case
an(a)
E
(0,
the
equations
x
accord-
Hence,
2(n+l)
comparison theorem, each of the functions
on
the
=
lim
E
otherwise
contradiction).
;;'-+ ) 2(n+l)
(0,
x
a
a
root in
(0,
; '+-
2(n+l) ),
which is
impossible.
Then,
< +oo.
e.
01
numbers k must have
An(ak) :! f (+oo) + (r(n + 1))2 < A, lim An(ak) +oo, i. e. we get k-oo
case
one
a
the
hand, by
+oo.
But
on
comparison theorem,
the other
contradiction.
hand,
it is
So,
as
in the
proved that
+oo.
=
+oo
It remains to prove that
half-line the
x
of Lemma
E
i.
opposite, some
that
(0, +oo).
E
a
proof
for all
is
monotonically increasing function
a
this, in
view of the
properties (a)-(c)
it suffices to prove that for any a,, a2
inequality u,,,(aj, x) :5 un(a21 X) 2(n+l) ) that there exist 0 < a, < a2 such that e. the
(0, 2(n+l) ; '-+ ). Let x, > 0 Un (a,, xi) Un (a2) xi). Let us
be the minimal
E
x
=
view of
of xj.
Suppose again
equality (11.4.8)
equation (11.4.6),
it is
an(a)
To prove
11.4.9,
'-+
(0,
neighborhood of
on
-=
with respect to
the
have
we
a
k
sufficiently large
contradiction.
a
as
_qk(x)u,,(ak, X)
+
+oo, i.
=
(because
+oo
--
a
x) satisfy
n,xx
to standard
as
'
2(n+l) )
U11
where 9k
+oo
--
lim
written for
u",x,,(a,, xi) n
>
>
Un
point
(a,, x)
x
=
we
xi,
have
u",xx(a2;X1)i
un(a21 X)
in
a
i.
==
Let X2
when
place. Suppose
Un(a2lx)
>
the
for
from this interval such
(a2 X) in a right halfun,x(al, xj) un,x(a27 Xl)- In An(al) < An(a2), hence, in view > Un
7
=
e.
we
get
a
contradiction.
right half-neighborhood
We also note that it follows from these arguments that
An(al)
the
0 < a, < a2
:
un(ai,x)
the contrary. Then
n
.
proved that u,,(al, x)
show that
takes
on
indicated in
un,x(al, xi)
of the >
So,
point
xi.
un,.'(a2) X1)
if
An(a2)X2
(n+l)
be the minimal value if there is
no
such
a
deriving inequality (11.4.9)
(xj, 2(n+l) such that Un(aj) x) Un(a27 X) or 1 point in (xi, 2(n+l) ). Repeating the procedure used x
E
with the
=
integration
over
the segment
[XI, X21
,
we
CHAPTER 2.
72
STATIONARY PROBLEMS
get: 272
I
0 >
un(al, x)u,,(a2, X)[f(U2(a,, x))
_
f(U2 (a2 X))
An (a,) + An (a2)] dx;
-
,
n
n
X1
in
the above arguments,
addition, by
An(a2).
Thus
get
we
a
function of the argument
increasing
Lemma 11.4.12 For any
the
and this
(0, 1) the
addition,
in
Suppose
E I
x
is
pair
for all the
u'
(xo)
n
'
0.
Therefore, only
(XlIX2); Un+l(XI)
E
[X I
i
X
2(n+2)
21
we
,
An+1
-
n
there should exist
following
0 < x,
coefficient
it suffices to prove
0.
=
case
of the
7
The solution
> 0.
n
An
=
a
xo E I
point
A and B
and also the
un+,(x) Un+I(X2)
integral
such that
can occur.
such that
Un(Xl)i Un+I(X2) Un(X2)7 Proceeding as when deriving (11.4.9) with
1-
X2
0 there exists
n
following inequalities
Proof. In view of Lemmas
for
So, >
a
integer
0
=
un(x) Un(X2)
if if
the segment
over
inequality
X2
i
0 >
Un+l(X)Un(X)[f(U
2 n
+1
(X))
_
f(U2 (X))
-
n
An+1
+
A,,]dx,
X1
where, Thus,
in the
as
get
we
a
proof
of Lemma
contradiction,
11.4.11,
and the
the strict
case
A is
inequality
takes
place
if
An+,
==
An
impossible.
un+-I(x) < u,(x) for all x E I. Observe that Un+I(X) < Un(X) for some 1 (because otherwise we would have u'( 2(n+2) ) 0). Further, un+,(x) :! ' Un(X) x E [0, n+1 (it is obvious visually on a picture). Let us prove that then
B. Let xo E I
for all Un+1
=
n
;kj lj
+ 1 consider
interval, the for each k
n+1
1). Change the function Un+1 on the segment Ik by the function v(x) X + Un+1 (y k-1 ) for x < 7 and to un+1 (Y x + n+1 k ) for x > Y. Repeat n+1
n
+
-
-
this
procedure
for each k
1, 2,
=
of the
properties (a)-(c) (see
I v (x) I
lies under the
Jv(x)j
such that
graph
Jun(x)l
G(, n(bn), Un(bn7 X))
!
(r(n + 1))'u'(bn7 X)
we
sufficiently large
would get that in
for all
F(u'(bni X)), (11.4.11) n
implies that dn
for all
-
n
(11.4.11)
sufficiently large
contradiction.
So,
numbers
numbers
Indeed,
n.
the n
< 20
if
we
suppose that this is not so, then
side is greater than the left-hand
right-hand and for all
x
satisfying jUn(bn,x)j
the uniform boundedness of the sequence
20,
i.
one
e.
a
jUn(bn7X)Jn=0'1,2....
is
=
proved.
jun(bni -)IL2(0,1) > I for all sufficiently large numbers n. h' (0) and, in view of 10 sin[7r(n + 1)x] and observe that u',x(bn) 0) hn(X) hn (1). We have: properties (a)-(c) from the proof of Lemma 11.4.9, u'n,,,(bn) 1) Let
us
show that
=
=
n
n
'
=
-h" n
=
/-tnhn7
X
G
(0, 1),
Set the
CHAPTER 2.
74
h,(O) where un
(11.4.6)
=
(7r (n
+
=
"+ -Wn
Wn(O) family
C,
constant
a
this
Multiplying
-
hn(x): x
y.w,,,
W,,(O)
=
=
n
f W,,(x)}
is
(0, 1),
E
W,,(l)
0,
=
n
uniformly bounded
family of functions fun (bn, X) ln>o and
J- n with
=
Wn(l)
=
of functions
boundedness of the
W.(x)
0
=
from the previous equation and equation
1))'. Therefore, we get w,,(x) u.,,(b,,, x)
for the functions
where the
h,(1)
=
STATIONARY PROBLEMS
Pn1
-
independent of
> 0
I n.
for
This
0), the positive solution is always unique (if it exists). in result of similar contains a error a on page 111. presented principal [83] proof ==
-
=--
Concerning the result from Section 2.4 on basis properties of the system of eigena nonlinear Sturm- Liouville-type problem, to the author's best knowledge
functions of this is
quite
a new
mention the
this
on
field and there
monograph by
subject and the
paper
system of eigenelements of ear
announced
a
by
the
same
results in this direction. We
no
[62] containing author [63] where
a
linear
problem,
(without
a
proof)
in
is
and
[5]
the
only
results
interesting
some
of the
completeness
equation, arising under small nonlin-
nonlinear operator
of
perturbations
almost
are
A.P. Makhmudov
The
proved.
Bary theorem
was
in the first
[6]. A more thorough discussion of [39]; the approach of this monograph [6]. The first proof of Theorem IIA.5 is
proved
in
questions around this theorem is contained in to this theorem differs from the
established in
fortunately, lems and
can
a
approach
in
in that paper the
be corrected. These corrections
results
corresponding
eigenfunctions
(without
[114]. However,
in
L2
are
[115-117].
in
is
a
negative
half-line
x
Here
>
0
we
constant.
over
have
Also,
a
This
approach
is
[119]
of
an
exploited
this paper shows that in
published
being
In
a
an
a
a
basis in
analog
nonlinear
general
in
attempt
to
the we
prob-
boundary-value problem
[1181 it is proved that the H'(0, 1) where s < so and
eigenvalue problem
use
which,
Similar
of the Fourier transform
[118]. However,
the
[119].
in
errors
basis for their systems of
a
[118],
proof of Theorem 11.4.5
Another natural way consists in aim.
in
eigenfunctions presented
contains essential
is considered. In the paper
system of its solutions, which is denumerable, is so
are
the property of
on
presented
spectral parameter)
proof
based
on
the
Bary
note that
the
theorem.
expansions (11.4.17) an
of the coefficients
on
is considered.
for this
example
from
properties b,',, (the (bn )n,m=0,1,2,... is upper triangular and all elements of its principal diagonal are nonzero) are insufficient even for the system of functions JUn}n=0,1,2.... to be complete in L2(0) 1)M
matrix
Chapter
3
Stability
of solutions
In this
chapter,
shall consider
we
unstable with respect
oo
-4
with
(for
waves
study the stability
to the distance p in the
respect
definitions of p and d
see
case
Introduction
or
As
waves.
of the KdVE and NLSE
of
solitary
are
as
Lebesgue
or
waves
vanishing
as
to distances of standard spaces of functions
Sobolev spaces and it is natural to
JxJ
stability of solitary
the
Introduction, usually solitary
it is noted in the
NLSE
questions of
of the KdVE and to d for the
Section
We also recall that
3.1).
case of the KdVE O(w, x wt) and u(x, t) e'wto(w, x) for the NLSE, a kink if 0' (w, x) = 0 for all x E R and a 0, xo is a point soliton-like solution if there is a unique xO E R such that 0' (LO, xo) of extremurn of O(Lo, x) as a function of the argument x and 0(-oo) 0(+oo).
we
named
solitary
a
wave
u(x, t),
where
u
(x, t)
=
in the
-
=
x
=
x
=
In the mathematical literature devoted to the
pioneer
paper
by
Benjamin [7]
T.B.
was
the
origin
stability
which initiated further
investigations
in the field. In this paper, the author has
like solutions
vanishing
as
x
--->
terminology
O(x -Lot) vanishing to each other in the can
as x -+ sense
can
oo
be
and
or
stability
a
the
the
stability of stability easily understood visually:
solution
of the distance p for
be not close to each other in the
Lebesgue
proved
the
f (u)
oo of the standard KdVE with
respect to the distance p; he called this like solution. This
of solitary waves, the
sense
u
(x, t)
some
form
if
a
numerous
of soliton-
t >
0, then clearly
soliton-
a
travelling
of the standard KdVE
with
u
=
of
wave
are
close
these functions
of distances of usual spaces of functions
Sobolev spaces, however, for this
t > 0
there is
a
translation
'T
=
r(t)
as
G R
u(x -r, t) and O(x wt) as functions of x are almost identical, i. e. the "forms of the graphs" of the functions O(x -wt) and u(x, t) "almost coincide". oo is stable, then for each t > 0 Thus, if a soliton-like solution vanishing as JxJ of of and the an arbitrary perturbed solution sufficiently the forms of its graph graph such that the
graphs
of
-
---->
close to it in the
sense
of the distance p "almost coincide". In Section 3.1
two sufficient conditions for the
solutions
vanishing
which
suppose to be
we
as
x
--+
stability
oo for
our
KdVE.
one-dimensional,
P.E. Zhidkov: LNM 1756, pp. 79 - 104, 2001 © Springer-Verlag Berlin Heidelberg 2001
we
consider
with respect to the distance p of soliton-like
i.
e.
Simultaneously
with N
=
1.
we
study
the NLSE
CHAPTER 3.
80
STABILITY OF SOLUTIONS
As it is noted in Section 2.1, the KdVE under natural behavior
Section 3.2 p under
oo of derivatives in
as x --+
have solitary
we
only
prove the
of
assumptions
In Section 3.3 as x --+
of
waves
we
oo. In two
In this
x
of solutions and the NLSE with N
types: these
stability
cases we
begin
the
I
can
=
soliton-like solutions and kinks. In
are
of kinks of the KdVE with respect to the distance
a
stability of solitary stability of
a
prove
waves
of the NLSE
nonvanishing
interesting type.
a new
of soliton-like solutions
section, the stability of soliton-like solutions vanishing
considered. We
on
general type.
consider
Stability
3.1
two
assumptions
with
the
defining
stability
as
x
-+
00 will be
of such solutions for the KdVE and
one-dimensional NLSE. Let
p(u, v)
=
inf
I Ju(-)
-
v(-
-
r)
u,
v
E
H1
TER
and
I Ju(-)
inf
d(u, v)
-
eiy v(--,r)jjj,
u,vEH',
'rER, yE[0,21r]
where H1 is the real space in the first prove that in each
First.
we
case
consider the
ut
remark that the
for the
f (.).
complex
in the second. One
can
easily
+
f (u)ux
+ uxxx
(X, 0)
=
for the KdVE:
uo
with
problem
following
111. 1.1 Let
f(.)
an
x, t C:
0,
=
R,
(x);
proof of the uniqueness of a
We also present the
Proposition
and
Cauchy problem
U
we
case
the greatest lower bound is achieved.
(111.1.2)
H 2-solution from Theorem 1.1.3 holds
arbitrary
twice
continuously
differentiable
result here.
be
a
twice
continuously differentiable function
and
u(.,t) be a H'-solution of the problem (M.1-1),(X.1-2) in an interval of time [0, a], a > 0 (uo E H 2 if a [0, a) or I 0). If there exists C > 0 such that < C for all t C- I, then there exists 6 > 0 such that the solution u(-, t) can be I Ju(-, t) 111 continued onto the interval [0, a + 8) (resp. there exists a H 2 -solution of the problem (111. 1. 1), (111. 1. 2) in an interval of time [0, 6), 6 > 0, if a 0). Proof. Let f (-) be a twice continuously differentiable function and u(., t) be a H 2_ solution of the problem (Ill. 1. 1), (111. 1. 2) in an interval of time I [0, a) or I [0, a], 0 can be considered by analogy). a > 0, bounded in IV: Jju(-,t)jjj < C (the case a let
I
=
=
=
=
=
=
=
Let
C,
>
0 be
a
constant, existing in view of the embedding of H' into C, such that
81
STABILITY OF SOLITON-LIKE SOLUTIONS
3.1.
lg(-)Ic
:5 Cl for all g
differentiable condition
E
(1.1.3),
so
that the
in I and due to the
Thus
[0, a
(., t) 6). El
is
ul
+
obviously,
since
of such
0 such that
>
H 2-solution of the
a
taken with
uniqueness
there exists b
Also, obviously
-
problem
Then
t).
JlgJJl < C and let fl(.) be a twice continuously Cl, 1 + Cl ] and satisfying f (.) for u E [-I
with
function, coinciding
H 2-solution ul (-,
global problem
H' obeying
(., t)
u
is
a
solution,
a
Jul (., t) Ic
f
==
f, has
unique
a
H 2-solution of the latter u
(-, t)
=
ul
(., t)
Cl for t E
< I +
for t E I.
[0, a
+
b).
in the interval of time
problem (111.1.1),(111.1.2)
00
Definition 111. 1.2 Let
f (-)
U C2 ((-n, n); R),
E
that in accordance with
so
Propo-
n=l
sition 111.1.1
H2-solution, O(w, -) E Hl.
for
and let Then
0(w,
x
wt),
-
problem (111.1.1),(111.1.2) has E
w
call this solution
we 2
H 2 the
E
any uo
be
R,
0 stable if for
p(uo(.), O(w, .)) of the problem (111. 1. 1), (111. 1.2) can be continued one has p(u(., t), 0(w, .)) < e for all t > 0.
such
E H
that, if uo
By analogy,
and
(111.1-3),(M.1-4) one
are
has
0
E >
proved
consider
P.L. Lions to the the KdVE with
stability
we
sumptions
c
investigation
f (u)
assume
=
that
Jul', v
E
>
0.
[2,4);
of the
of the space
H'.
=
for
any uo
H' satisfy-
E
stability When the
half-line
t > 0 and
the
for
problem all t > 0
uniqueness of this solution u(x, t)
concentration-compactness method of of soliton-like solutions. We consider
formulating
requirement 111. 1. 1
However,
problem (111.1.1),(111.1.2)
H')
t > 0 and
R,
E
0 such that
of the
Proposition
differentiable function.
well-posedness
u(x, t)
f(JU12)U in equation (111.1.3) e'wtO(x) be a soliton-like U(x, t) Then, we call this solution V(x,t)
local existence and
of the
v
half-line
corresponding H'-solution u(x, t) of
application
an
of Theorem 1. 1. 3 and
continuously
sense
(the L2.4).
e
of
uo(x).
Section 1.2 and let
condition
solution
=
0,
=
any
corresponding
onto the entire
Cauchy problem for
iUt + AU +
satisfy
then the
6,
unique local
a
soliton-like solution
a
if
in
v
the
according one
a
proves
suitable
for other values of the parameter
on
the
connected with
as-
following
> 2 is
to which or
E
f (.)
is
a
twice
supposes the local
(for example, in (0, 4), then all the
sense v
result
STABILITY OF SOLUTIONS
CHAPTER 3.
82
arguments from the proof of the theorem below hold.
Theorem 111.1.4 Let
O(x,t) from ftn}n=1,2,3....
fix
us
the
Kd VE
and
a
A > 0.
arbitrary
family (11.1.3)
R+
C
an
[2,4). Then, for any A > 0 the (Iff. 1. 1) from the family (11. 1. 3) is stable.
where
Jul"
=
O(x, t) of the
soliton-like solution
Proof. Let
f(u)
E
v
Then, there
is not stable.
ju(n)}n=1,2,3....
sequence
that the
Suppose
0
exist
>
E
from H2such that
a
sequence
of the
0
_
0
and
oo
0,
P(U(n), 01t=0)
P(Un(*i tn)i Olt=tn) > e where Un(Xi t) are H2-solutions (n) problem (111.1.1),(111.1.2) with uo given by Theorem 1.1.3. uo A > 0. Consider the minimization problem Let lo(.,t)122 as n -+
solution
corresponding
Cauchy
=
=
inf
IA
E(u),
uEHI, JU12=A>o 2
where the functional has
solution and
a
and
> 0
w
E(u)
is defined in Section 2.3.
clearly every
boundary
to Theorem
According
its solution satisfies the
following equation
11.3.1,
with
it
some
conditions I
U" + V
(the parameter the above
this
problem
we
01
problem
11.3.1,
A and
n
a
=
0 (up
Further,
since
10, 122 : 102 122
for any
to
with different values of the parameter a
--+
--+
our
sequence of solutions
A
minimizing
=
vn
as n --+ oo
-+
oo7 i.
e.
a
Aun (-,tn)
JU.(*,tn)12 and E(Un('7
results
-+
contradiction.
in the papers on
the
0))
C R such that n
oo.
to Theorem 1.1.3
according
E(un(-, tn))j
=
by
stability
Since
v,,(.
+
y.,,)
-->
lun (-,tn) 12
proofs
based
on
1,
111.1.4
T. Cazenave and P.L. Lions
with
problem,
we
111.1.4 is
time, Theorem
it is clear that
get
JUn(.,0)122 fvn},,=, 2,3....
therefore there exists
converges in H' to
A
Thus, Theorem
Remark 111-1.5 For the first
fact,
fUn}n=1,2,3,... of the problem
.Since
sequence for the above minimization
jYrjn=1,2,3.... 0 as Hence, p(vn, 0)
In
family (11. 1 -3)
2.1,
of Section
2.1, beginning translations, positive solution 0 that
0.
>
it is shown in Section
as
As at the
up to
A,
0
translation). Therefore, according minimizing sequence fWn}n=1,2,3.... of our minimization 0 in H' as sequence fYn}n=1,2,3.... C R such that wn(- + y,,)
sequence
n --+
some
=
for any
take
we
JUn(*) tn) 122 a
otherwise,
solutions).
unique,
a
U(00)
LOU,
because
positive
nontrivial
from the
02 =
there exists
Return to
a
no
=
oo.
For any
is
and
get that A,
to Theorem
n
has
jul"u
family (11.1.3) with
to the
two functions
--+
must be
boundary-value problem has
belongs A,
w
+ 1
P (Un ('I
0
tn)
as
n
oo.
--+
0
7
as
proved.0
was
[23]
proved
and
by
in the paper
P.L. Lions
[1011. [57,58] the
the concentration- compactness method
for
83
STABILITY OF SOLITON-LIKE SOLUTIONS
3.1.
essentially
NLSE of the
a
Now
ishing The
applying
we
jxj
as
in these papers. Here
of
possibility
which
oo
only
of the
problem
of the
from the
in the
(E0
has been often denoted
by Q.
physical
We also
the
particular,
a
spatial variable,
is
of
stability
the
example
an
solitary
waves.
of soliton-like solutions
stability
van-
close to necessary.
stability
literature in which the
conser-
used in the condition of the
KdVE)
of the
case
on
in
to illustrate with
sufficient conditions of the
gives
vation law P of the NLSE
stability
"Q- criterion"
"Q-criterion" originates
name
wanted
we
this method to the
consider the
--+
considered:
are
admitting coefficients depending
multidimensional NLSE,
investigated
kind
general
more
here the conservation law Eo
rename
U
of the KdVE
P and recall
by
in the
that,
the
case on
KdVE,
7(u)
=
f f (p)dp
and
0 U
F(u)
=
f 7(p)dp. 0
f(JU12 )u be a continuously differentiable function of the comI (f(u) be twice continuously differNLSE (111. 1. 3) with N
Theorem 111.1.6 Let
plex argument
u
for the
=
'r
f f (s)ds
F(r)
Let also
KdVE).
entiable for the
and let there exist
R and b > 0
wo E
0
such that
f (0)
-
wo
0 is
then this soliton-like solution
satisfied,
U(x, t)
is
W=WO
stable.
Remark 111.1.7 A similar statement is obtained in the paper like solutions with
positive
functions
author of this paper notes, his For
cases.
example,
complete. As
(U(x, t)
=
the paper
(N
=
we
O(w, x [35], 0
1,2).
in the
know,
wt)
for the
-
proof
case
for any
w
0
is
a
multidimensional NLSE.
incomplete
N > 1 and
a
where
a
O(x)
=
JuIv
solitary > 0
>
as
the
I except for several
the
wave
for soliton-
[92]
However,
proof from [92]
U(x, t)
and, according
=
is
e'wto(w, x)
to results of
from RN, if p E
(0, N'2) (N > 2) and p > 0 equation (11.0.1). Substituting O(x) WPV(W2X),
point
satisfies
when N
f(X, JU12)
> 0 there exists
KdVE),
is radial about
The function
0 of
=
84
we
CRAPTER 3. find in the
case
of the NLSE: AV
and
tion of the
have
P(U)
=
2
f
2
Wp
v
=
0,
Vjj"j_"
the
uniqueness of
Hence, by
problem (11.0.1),(11.0.3) 2.-E
we
IVIPV
+
V
-
for the KdVE.
by analogy
STABILITY OF SOLUTIONS
(y)dy
=
0
a
positive radial solu-
mentioned in Additional remarks to
where
v(.)
is
fixed function.
a
Thus,
Chapter 2, 0 if
>
dw
RN
0 < p < -1. N
[92] (for case
N
According to Theorem 111.1.6 (for N 1) and results from the the solution > 1) U(x, t) is stable under this condition. By analogy, =
of the KdVE with
what
f (u)
=
I u IP
the condition dp(o) dw
follows,
we
shall show that the
similar result
on
the
Proof of Theorem 111.1.6.
f (0)
0
=
and, respectively,
if necessary, where that
11hl 121
u(x, t)
> 4
place
making
the
change
One
can
(0, 4).
for the NLSE if p
proved
of variables
solution of the NLSE
a
f [lh'(x) 12 +wolh(x) 12 ]dx.
=
takes
for the KdVE is
We first consider the NLSE. We
wo > 0
is
instability
when p
instability
0 is satisfied if p E
>
(111. 1.3).
prove that the
in
Here
>
In
'. A
N
[15].
can
v(x, t)
paper
in the
=
we
accept that 6- f (O)tu(x, t) also accept
greatest lower bound
-00
in the
expression
remark that u
for
d( U, u)
generally
in the form
is achieved at
and 7
-r
e-'(-/W+w0t)u(-
real). Differentiating
the
+
are
r(t), t)
T(t)
E R and
(we
unique). 0 + h(x, t) where h(x, t) v + iw (v, w with to and T d(U, u) respect 7, we get
are
=
=
expression for
7(t)
E R
some T
We represent
not
a
perturbed solution =
00
I V[f(02)
+
202f/(02)] O'dx
=
X
0,
(111.1.5)
00
00
j Wof(02 )dx
0.
=
(111.1-6)
00
Further, AE+
WO
AP
=
2
where
a(s)
o(s) f(02).
=
d2
dx! + W0
E(u)-E(O)+
_
as
s
--
Wo
2
[P(O+h)-P(O)]
+0 and
Lemma 111. 1.8 There exists C
d2
L+
>
dX2
>
1f(L+v,v)+(L_w,w)}+a(jjhj 12)
2
+
0 such that
W0
-
[f(02) +202f/(02)]
(L- w, w)
CIIWI12 for
all
W
E
H'
satisfying (111.1.6). Proof. Let
w
of the operator L_
=
ao
+ wi- where
corresponding
(L-w,w)
(0, wi-)
to the
=
=
0.
Then,
eigenvalue A,
=
since
0
0,
have
we
(L_w.L,w_j-) ! A2 JU,_L1221
is the
eigenfunction
STABILITY OF SOLITON-LIKE SOLUTIONS
3.1.
85
1 2 is positive
2 is the greatest lower bound of the positive spectrum of L-;
where
because,
since is
and, hence,
is
0
an
the
positive function,
a
isolated
corresponding eigenvalue A,
of the spectrum of the operator L-
point
=
(on
0 is minimal
this
subject,
[28]). Then, by (111.1.6):
see
I 02f(02 )dx +IW_L0f(02)dX
a
00
00
Since
0.
=
f 02f(02 )dx=f (012 -00
W002 )dx
+
0,
>
get:
we
-00
Jal hence
IW12
C2
>
0
sup
I f(02) 1)
!5
of
independent
E
w
H'.
CIW112i ! C21 W2 12 with
(111-1.8) implies (L-w,w)
and therefore
C11WI12,
:5
For k > 0
independent
of
w
we
have
some
(M
X
00
I
(L- w, w)
1
(1W )k-+1 + 1
112
+
2
1
W01W122)
W
2f(02 )dx
00
I
k +
2
i-T-1 OW/12
W01W12). 2
+
Thus, k
(L-w, w) ! for k k
>
>
0 the
small and
sufficiently
0
1
expression
k+1
k + 1
IIWI121
of w, because for
independent
(IW112+WOIW12)_ f W2f(02 )dx 2 2
a
sufficiently
small
is not smaller than
-00
00
C2
k
2
k + 1
IW12
I W2f(02 )dx
-
k + I
>
(C2 k +
k _
1
k + 1
M) IW12 2
>
0.
-
-00
Lemma 111.1.8 is
In what
proved.0
follows,
we use
the condition
(V, 0)
=
(111.1.9)
0.
Lemma 111.1.9 There exists C > 0 such that
satisfying (111.1.5) Proof. Since value
2
=
0 and
there exists this
an
eigenvalue
and
all
v
E
H'
(111.1.9).
clearly 0' 0'
(L+v, v) >CIIVI12 for
has
is
an
precisely
eigenfunction one
eigenfunction g1(x)
> 0 with
is minimal. Lot gi >
of
L+
with the
corresponding eigen-
root, A2 is the second eigenvalue of L+
0,
92
=
a
corresponding eigenvalue A,
mo'
be
eigenfunctions
so
that
< 0 and
of the operator
86
L+
o,',,
CHAPTER 3.
normalized in L2 and let L_L be the
b92
ag, +
=
W
01,
+
v
kgl
=
+
192
+
(L+v, v) It follows from the
V I
0 is
>
Alk + (L+v
spectral theorem (see [28])
of
independent
0 _L, 1
,
orthogonal
vj-).
Let
(III.I.10)
that
C1 jvj_122
(III.
Further, equality (11.0.1) implies
v.
to gi, 92.
Lj-. Then,
vi- E
2
=
(L+v -L, v , ) where C,
of L2
subspace
where
STABILITY OF SOLUTIONS
that
-L+O".
Therefore, by (111.1.9)
v)
0
=
Hence, using the Schwartz inequality,
Alak + (L+0 i,
we
v i
obtain
i
(L+v,,Vj_)2(L+0,,0j-) Using
now
conditions of the
2
>
theorem,
I (L+ v-L, oi-) we
A, I a
k 1.
find
d
P(O)
0
0.
=
(111. 1. 14) implies:
we
STABILITY OF SOLITON-LIKE SOLUTIONS
3.1.
Proceeding further as we
87
proof of Lemma 111. 1. 8, from the last inequality
at the end of the
get:
(L+v, v) and Lemma 111.1.9 is
proved.0
Lemma III.1.10
I lh(., t) Ill
Proof. We have for
is
a
::
continuous
arbitrary tj
1 lu(-, ti)
-
ei('y(t1)+0j0t1)0(*
-
e
i('Y(t2)+WOt2)
0(.
function of t.
IIh(-,t2)II1
-
-
7(tl))Ill
-
-
7(t2))Ill
-
I IU(i t1)
and t2:
Ilh(.,tl)lll IIU(',tl)
C
CIIvII1,
JIU(*)t2)
-
JIU(')t2)
-
_
U(* t2)1 11 7
e'(^f(t2)+WOt2)0(. + 7-(t2))Ill ei(Y(t2)+WOt2)0(.
, _
T
-(t2))Ill
C2(lllmhj-+a20l, 2 1
where lim S
+0
0. 8
AE +
WO
Hence, there
AP ->
2
m
> 0
(73 (1 lIm h-L + a201 121
for all functions h
sufficiently
the condition
0,
coefficients a,
(111.1.16) satisfying
I lRe h-LI 11).
Also, clearly AE +
WO
2
AP
0,
(111.1.17)
88
CHAPTER 3.
for all h
small in H'. For
sufficiently
hood of
06 be the
neighbor-
open
in H' of the kind:
zero
06
Let
6 > 0 let
arbitrary
an
STABILITY OF SOLUTIONS
h
=
>
E
H"
m-1 be
a
jail
:
I IIrn
+
h
IlIm
sufficiently large
f lai(t)l
sup
1-8, 2
+0. Suppose this is
+0
-->
IlRe
oo, such that either
as n -+
h In
tn) 112
2 c2for
>
that for all
there exist
right. Then,
of solutions of the
u.,&, t)
sequence
not
some
tn
a
(sufficiently small)
problem (111-1.3),(111.1.4)
aln(tn) >
0,
c or
> 0
and
06,,, hn(', 0) I IIrn hi-n( -, tn)+a2n(tn)O(*7tn )112+ 1
1, 2,3,
n
C
with
....
E
First of
all, (111. 1.15) implies IIImhj_n(*it)+a2n(t)0 t) Ill2+
sufficiently large laln(t)l < E and IlIm hj-n(.,t) + a2n(t)O ( t 1112 + IlRe hj-n(., t)112I < _C2f2 because if lain(t)l IlRe h n(', t) 112I < V E2, then by (111.1-15) lain(t)l ! -6 +C562 which is a contradiction 2 n
wehave
euntil
=
_
-
1
-
because
tn
>
0,
n
we
arbitrary
C7,8,n2
-+
(111-1-18)
exists. These relations also
to the soliton-like solution
02,E2.
n:
>
-
contradiction,
u(., t)
=
get
JIU(*it)lll taking place
+
(111. 1.16), taking a sufficiently large
to
independent
[AE + Thus,
to
m[IIIm hln(*,tn) +a2n(tn)0(',tn)II1
[AE + for all
Hence, according
IlIm h1n(',tn)+a2n(tn)0(*)tn) 112
and
lal,,(tn)l
small
small.
1, 2, 3,..., such that
=
lain(tn)l Then,
0 is
>
c
yield
a
0
we
get
as n --+ oo.
are
proved for all
priori
0)
problem (111. 1. 3), (111. 1.4) sufficiently
point
for those t > 0 for which these solutions exist.
t for
estimates
0 in the
close
sense
of the distance d and
Hence, according
to Theorem 1.2.4 and
t
=
STABILITY OF SOLITON-LIKE SOLUTIONS
3. 1.
any of these solutions
proved statements, the entire half-line t 111.1.6 is
us now
of
change
u(x, t)
case
global (it
is
be continued onto
can
of the NLSE the statement of Theorem
proved.
Let ate
for the
0). Thus,
>
u(., t)
89
consider the that
variables,
in the form
u(x
+ r,
of the KdVE. We
case
f (0)
t)
=
0 and wo > 0.
O(wo, x)
=
assume,
Representing where
h(x, t),
+
can
making a
ff(x, t)
the soliton-like solution under consideration and the parameter
appropri-
perturbed =
r
an
=
solution
O(Wo, x Wot) is r (t) is chosen to -
minimize
U, x
we
(.
+
r,t)
_
01 (Wo'.)122 +WOIU(.
+
x
7-,t)
_
opo'.)122)
get the constraint 00
I f (O(wo, x))O'(wo, x)h(x, t)dx
=
(111.1.19)
0.
x
00
Also,
as
in the Wo
AE,
of the NLSE
case
+
AP
=
2
El (u(., t))
(Lh, h)
>
where the
0
case
of the
+0 and L
as s -->
NLSE,
we
=
-4 -
dX2
-
flo(wo, .))]
!
7(1 IhI 11),
+
+
[P(u(., t))
2
Loo
-
f (O(wo, -)). Proceeding further
as
in
get the estimate
(Lh,h) for all h E H1
Wo
El (O(wo, .)) +
-
satisfying (111.1.19)
C911hl 121
>
and the condition
00
j
h (x) 0 (Loo,
x) dx
=
0.
00
The end of the
111.1.6 is
proof
of the
stability
Remark III.1.11 Under the
NLSE and KdVE the d
< 0.
In the
in the papers
simplify from
only
our
if 0
instability
case
case
repeats those for the NLSE. Theorem
[40,84].
< P
p > 0 for N a
radial
=
for the
for the NLSE
IUI,'. According
0 has
=
[15];
a
of the
cases
with respect to the above distances takes
of the KdVE it is
consideration,
Chapter 2,
these
in this
proved.0
place if NLSE,
and,
to
to results
positive solution
if and
1, 2); in addition, if p satisfies
positive solution 0. According
to
90
CHAPTER 3.
Remark
111.1.7,
we
have to prove the 4
( N4, N-2 ) (incomplete) proof.
the distance d for p E a
formal Let
w
us
positive
where
a
positive radial
constants
solution
x
in Section 2.1.
as
e"OtO(wo, x) a
of the NLSE.
function h
complex
jjhjjj<e,
(p
>
of the
0
e""O(x) with respect to 1, 2). We present only
solutions 4
N
for N
=
First of all, for a given number problem (11-0-1),(11.0.3) there exist
and b such that
a
10(x)l for all
N > 3
the notation from Section 1.3.
use
0 and
wo >
=
instability of
STABILITY OF SOLUTIONS
=
Then, z(t)
Further,
h,
IVO(x)l
+
+
ih2
0
C
U(x, t)
there exists
H' such that the following inequalities take place:
E
jY0+h(0)-Y95(0)j1E-1,
and
(111.1.19) where YO+h and ZO+h
are
e""o O(wo,
of the
we
x) + h (x, t)
choose
can
hand side of
decreasing
a
are
0.
>
there exists T
ZO+h(t)
determined.
>
0 such that
to the solution
with uo
0 + h. Thus, 0 the
it is clear that
Further,
is
function of t for all t
an increasing Therefore, inequality (1.2.19)
the function
Hence,
corresponding
small in H' such that at t
arbitrary
is not smaller than 1.
function and
z
Cauchy problem (111. 1. 3), (111. 1.4)
function h
(1.2.19)
these functions values of t
values of the functions y and
Z.O+h(t)
Zq5+h(t)
---
+oo
takes
right-
YO+h(t)
is
a
>
0 for which
place
for all these
cannot be continued for all t > 0 and
as
t
--4
T
-
0.
Hence,
IVU12
SUP
=
+00
tE[o,T]
where
u(-, t)
e"00(wo, x)
=
of the NLSE is
Stability
3.2 In this kinks
section,
are
10(w, x) 1
we
x
assumptions
the
only
and the
KdVE,
stability
like
e'woto(wo, X)
X
are
of the
there is
=
"almost
always"
general type
a common
exploit this idea,
(h, Lh) (see
of, kinks of the KdVE. We recall that
O(w, x wt) satisfying the conditions 0' (W, x) : 0 and O(w, oo) 0+. As it is noted in the Introduction, the
We set
J.F. Perez and W.F. Wreszinski we
of the solution
-
stable. It
on
in what
means
the function
main idea in the
for various "soliton" equations; this idea first
Here
instability
of kinks for the KdVE
waves
E R.
solutions of this kind natural
h(x, t)
shall prove the
travelling
:5 C,
+
proved. 0
[42]
where
a
that
analysis
appeared semilinear
they
are
f (u). Although of the
we
stability
in the paper wave
stable under
by
equation
consider
of kinks
D.B.
Henry,
is considered.
too. It is used for estimates from below of the functionals
follows).
Consider the KdVE considered in Section 2. 1:
Conditions for
a
kink
O(w,
x
-
providing
the existence of kinks
wt) satisfying equation
are
(11. 1. 1) and
STABILITY OF KINKS FOR THE KDVE
3.2.
the conditions
0(oo)
ing conditions
are
0
=
it is sufficient and necessary that the follow-
exist,
to
91
.0
satisfied
(hereAo)
f f(s)ds, fj(0)
=
=
7(0)
-
wO
+
wo-
and
00
f fi(s)ds):
F1(0)
0-
A:
fi (0-)
fl (o+)
B:
F, (0-)
F, (o+)
0;
0;
C: F, (0) < 0 for all
We also
(0-, 0+).
G
require -W
Clearly,
condition
(111.2.1) provides
I O(W, X)
-
Without the loss of and a
(111.2. 1)
we
suitable result
on
+
A0)
< 0.
(111.2.1)
the estimates
I Ox'(W, X) 1
generality
shall show the
we
!5 C1 e-C2 1XI)
C1 C2
kink
uniqueness
with conditions
on
the
>
O(w,
of
x
0-
0- Under conditions A-C -
wt).
solution
a
>
i
accept that 0+
stability of the
the existence and
problem (111. 1. 1),(Ill. 1.2) is the
0I
+
For this aim
u(x, t)
of the
infinity u(00, t)
we
need
Cauchy
This result
following. Theorem 111.2.1 Let the assumptions A-C and
(111.2. 1) be valid, f (-) be a twice function uo(-) be such that u0(-) O(w, -) E Then, there exist a half-interval (0, a) and a unique solution u(x, t) of the problem (111.1.2) such that u(-, t) O(w, -) E CQ0, a); H 2) n C1([0, a); H-'). For any
continuously differentiable function H2
.
and
a
-
-
of
these solutions the
quantity
IM-1 0)
2IU (X, t) 2
=
-
x
F, (u (x, t))
dx
00
does not
depend
on
t, i.
e.
the
the above solution exists
on a
that
C
of
I Ju(-, t)
-
0(w, -)111
this solution onto
a
functional I(.)
half-interval [0, a), all t E
Proposition
a
conservation law. In
a
>
0, and there
[0, a), then there half-interval [0, a + 6), 6 > 0.
0 such
proof
continuation
of Theorem
STABILITY OF SOLUTIONS
CHAPTER 3.
92
Remark 111.2.2 To get
it suffices to write the
111.2.1, nition
careful definition of solutions
a
equation
for the difference
u
from Theorem
u(x, t)
and formulate
0
-
a
defi-
to I.I.I.
analogous
Remark 111.2.3 Since by construction
quantity I(u(-, t))
IF1 (U)I
C(U
0 and for any t > 0 the inequality Pq(U(', t), O(W, -)) < E takes place.
H
and
(111. 1. 1),
Remark 111.2.5 One the
again
stability
easily
can
that the
see
stability from Theorem
Proof of Theorem 111.2.4. We first prove the
J(U) where T
-
a(s) o(s) as s Lot) + h(x, t) where
easily
=
-*
T
_
CP2(U, 0)
I(0)
T(t)
following
-a
q
+0 and C =
>
0 is
estimate:
(P2(U' 0)),
(111.2.2)
.
independent
C- R is chosen for
prove the existence of such
of
u.
Pq(U, 0)
Let
u(x, t)
=
to be minimal
O(W, x (one can -
T). Then,
AI 00
=
I(U)
-
1(0)
00
Ih,2
2
111.2.4 is
of the form.
+
[w
-
f (0)]h 2jdx-
-00
-1
I If (0
2
+
Oh)
-
f (0) 1 h
2
1
dx
=
-
2
Ij(h) + a(P2q (U, 0)),
-00
(111.2-3) "0
where 0
O(x, t)
E
(0, 1),
a(s)
=
o(s)
as
s
--*
+0 and
II(h)
f -00
L
d2 =
dX2
+
hLhdx with
STABILITY OF KINKS FOR THE KDVE
3.2.
Now
we
(for details, set [a, +oo)
the
use
where
symmetric ordinary differential operators
of
spectral theory
The continuous spectrum of the operator L coincides with the
[28)).
see
93
a
w
=
-
maxjf (0+), f (0-)}
Since
0.
>
0'(x)
>
0 and
LO'
=
0, the function 0' is the first (positive) eigenfunction of the operator L with the minimal
corresponding
if it exists. We take b
Let h
ILO'
=
eigenvalue A,
A2 if A2 exists and b
=
g)
+ g where
0. We
=
Ij(h) Further, differentiating
Hence, the second eigenvalue A2
0.
=
2
p q (U,
a
=
in the
opposite
is
obviously have
bjg 122'
::f
positive
case.
with respect to
(111.2.4)
-r
at the
point of minimum,
we
find
00
I [q
Lo
-
+
f (0)]O'hdx
=
0.
00
Substituting
h
=
po'
+ g into this
equality, P '5
which
with
together
last,
we
>
in the
and
, ,
0.
C3
>
(111.2.5)
0, follows from
(111.2.3)
As in Lemma
III.1.10,
inequality (111.2.2) easily
(111.2.5). continuity
t. Take
and 0
an
if(a2)t 0 where aoe ,
4)(t) u(x) 54 =
the
1) non-vanishing
shall prove
we
solutions
studying
=
on
0, and kinks. As a new
ao > 0 is
one-
oo. These solutions we
already
interesting type. a
We
parameter.
(this is valid, for example, if X' is close in such to constant a nonzero as 4D(t) with a fixed t E R). u(.) sufficiently Then, one can easily verify that there exist a unique real-valued function a(.) E X' and a function w(.), absolutely continuous in any finite interval and unique up to c
E
adding 27rm,
m
=
E
0, 1, 2,...
and that ao +
a(x)
=
lu(x) I
u'(x) have
(ao + a(-))w(-)
> 0
=
(ao
for all
[a(x)
=
E R
it, such that
to
u(x)
x
+
a(x))e'lc+w(x)]
+
x
i(ao
E
+
R;
in
addition,
(111-3-1) due to the relation
a(x))w'(x)]e'1c+w(x)1,
L2 In particular, if 0
(111-3.2)
a(.) < C2 < +00 for all x E I I a(.) I I, is sufficiently small, then W'(.) E L2 Conversely, if for a complex-valued function u(.) there exist real-valued functions a(-) E X1 and w(.), absolutely continuous in any finite interval, such that (ao + a(.))w'(-) E L27 ao + a(x) > 0 for all x E R and that (111.3.1) with some c E R takes place, then u(-) E X1 and u(x) : 0 for all x E R. The result on the stability of solutions 4b(t) we we
R, that
for
E
.
example occurs
consider here is the
< cl < ao +
if
-
following.
Theorem 111.3.1 Let N
f(JU12 )u be a twice continuously differentiable junction of the complex argument u, f (.) be a real-valued function and f'(ao2) < 0 for Let bo > 0 be such that la(-)Ic < E9- if a(.) (-= H' and Ila(.)Ill < 6o. some ao > 0. 2 if(aDt of the NLSE (111.1.3) is stable in the Then, the solution 4D(t) aoe following sense: for an arbitrary 6 E (0, bo) there exists 8 E (0, 6o) such that, if uo(-) E X1, uo(x) : 0 and uo(x) (ao + !T(x))ewW where u(x) luo(x) I ao E H1, I IZT(.) 111 < 8 and IW'(') 12 < 6, then the corresponding X'-solution u(x, t) of the Cauchy problem (III.1.3),(M.1.4) given by Theorem L2.10 can be continued onto the entire half-line t > 0, u(x, t) : - 0 for all x E R, t > 0 and for any fixed t > 0 the functions a(x, t) andw(x,t) in the representation =
1,
=
=
=
u(x, t) satisfy thefollowing: a(-,t)
E
=
(ao
+
-
a(x, t))e'U(-5')+-(x,t)1
H', Ila(.,t)lll
ClIal 12I
,
a
function of t E
ao+a(x,t)
>
0 and
H' and the function
[0, to). Also,
in view
exists C > 0 such that
E
H1.
(111.3.3)
STABILITY OF SOLUTIONS OF THE NLSE NONVANISHING AS
3.3.
Take
JXJ
oo97
--+
(0, bo) and let H(s) Cs' + a(s') where C > 0 is the constant for (111.3.3). Then, there exists 61 E (0, c) such that H(s) > CS2 2 (0, 61]. By the above arguments, there exists 82 E (0, 61) such that arbitrary
an
E
6
=
from the estimate any
E
s
C
H(jja(-,t)jj&5 M(u) if
Ila(-,0)jjj
82 and JW,1 (*) 0) 12
0 and all x E R, where u(x, t) is a for
a
-
t > 0.
=
=
-
=
-
=
=
=
X
X'-solution of the problem
(M.1.3),(IIIJA),
Theorem 111.3.7 Let N tion
the
of
complex argument kink
corresponding
V(x, t)
=
1, f(JU12 )u be
=
and let the
u
e'w_t0(x) of the
then
a
we can
twice
introduce
Lo(x, t).
continuously differentiable func-
assumptions (a)-(e) be valid. Then, the NLSE
(111. 1. 3)
is stable in the
following
sense:
for
if uo
X1, luo(-)l 0(-) E H1, Ila(., O)l 11 < 6 and ILO,1(*) 0) 12 < 6, then the correspondu(x, t) of the problem (Iff. 1. 3), (111.1.4) is global (it can be continued
G
sufficiently
any
small
c
>
0 there exists
a
sufficiently
small 6 > 0 such that
-
ing X1 -solution onto the entire
Ila(.,t)lll
and
c
0)
0
one
has
I U(-, t)
0(-)
E
H1,
6-
Proof. Consider the functional 00
J 2 1UX(X)12
M(u)
_
U(JU(X)12)
+
WIU(X)12 + D, 2
dx,
00
U702 -i
where D,
-
+
U(02 ).
interval
of
time
can
easily verify that for the quantity M(u(.))
E
to be
X' it is sufficient and necessary that lu(.)I-O(-) E L2be a X'-solution of the problem (111-14,(111.1.4) in
u(-) u(x,t) [0, to) and let lu(-, 0) 1
Lemma 111.3.8 Let an
One
-
determinedfor afunction
-
0(-)
E
L2. Then, lu(-, t) I
-
0(-)
E
L2 for
3.3.
STABILITY OF SOL UTIONS OF THE NLSE NONVANISHING AS
[0, to)
all t E
so
quantity M(u(., t))
that the
is determined
for all
IX I
--+
oo99
[0, to) and,
t E
in
addition, independent of t.
f(JU12)u be a five times continuously differentiable function, uo E X' and u(x,t) be the corresponding X4_solution of the problem (Ill.1.3),(IIIAA). dM(u(-,t)) 0, hence, the quantity M(u(., t)) Then, the direct calculation shows that dt Proof. Let first
=
of t E
is determined and
independent
of the lemma
passage to the limit
and
uo
by
a
For
uo E
AM
X' nonequal
:--
M(UO)
-
to
zero
M(O)
statement
sequence of smoothed functions
a
we
f( JUI 2)U
ones. 0
luo(-)l
and such that
M(V)
-*::--
get the
For
over a
non-smoothed
to the
converging
X'-solution
[0, to).
M(O)
-
=
-
0(-)
E
H'
we
have
(111-3.4)
M1 + M2 + M31
where 00
00
Ila
M, 2
2 X
(0
+
_
f(02)
-
202f/(02 ))a 2}dx,
I W12 (0
M2
+
X
2
a)2 dx,
-00
-00
00
I
M3 2
where 0
ja21f(02)
=
O(x, t)
E
+
202fl(02)
f ((0 + Oa)2)
-
2(0 + Oa)2f/((o + Oa)2 )dx (111.3.5)
(0, 1).
Lemma 111-3.9 There exists C, g E
-
0 such that
>
Mi(g)
CIJgJ22 for
all real-valued
H1 satisfying the condition
(91 01)
=
(111.3.6)
0.
-0 + f(02) + q(x) _Zj + f(02 ) + 202 fl(02 ). By condition (b) we have q < 0. 202f/(02) Let q The continuous spectrum of the operator L fills the half-line [b, +00) where b minjq_; q+} > 0 (see [28]). Further, it follows from equation (111.1.3) that 0'(x) is an
Proof. Consider the operator
LW
=
-w"
-
&) o
where
=
=
.
eigenfunction since
0'(x)
tor L.
is of constant
Hence,
all 9 E H'
of the operator L with the
I I Ju(-
-r,
is the smallest and isolated
it follows from the
satisfying (111.3.6),
The number -
sign, Ao
ro
was
corresponding eigenvalue A0
spectral
defined
as
a
point
0; in addition,
eigenvalue
theorem that M, (g)
and Lemma 111.3.9 is
=
=
(g, Lg)
of the opera>
C1 Ig 1 22 for
proved.0
of minimum of the function
Therefore,
t)
"0
0
2
=
(R (T))
1
'r='ro I a(x,t)o'(x)[q 2
00
ay +
f(02)
+
202f/(02 )]dx.
R('r)
STABILITY OF SOLUTIONS
CHAPTER 3.
100
We take
sup
jZj
-
f (0'(x))
-
20'(x)f'(O'(x)) I
+ 1 and
xER
10112 1 01(V
K
+
f(02)
+
202ff(02)) 12
00
f
+
f(02)
+
202fl(02))dX
-00
Lemma 111.3.10 g E
C2Ig 12, 2
MI(g)
where
C2
Cj(1
=
+
K) -2, for
all real-valued
H' sat,sfying the condition W
I g(x)o'(x)(q
Zj
-
+
f (02(x))
+
202(X)f1(02(X)))dX
=
(111.3.7)
0.
00
Proof. g
=
ao'+
Represent
o where
arbitrary function
an
(0', p)
0. Then
=
M, (g)
9 E
H1 satisfying
CIJW12. 2
M, (W)
=
(111.3.7)
in the form
We get from condition
(111.3.7): 00
00
a
j 0/2rq
_
Zj +
f(02)
+j W01[q
202f/(02 )]dx
+
_
ZU +
f(02)
+
202f/(02 )Idx
=
0,
00
-00
hence, co
f 001[-q I a 1101 1 2
=--
101 12
+
f(02)
202f1(02)]dX
+
< oo
f 012[-q
+
f(02)
+
202f1(02)]dX
-00
I ot [-t7
I W 1 2 10112
0, we get dimensional Gaussian Now
we
span
=
I el,
.
-
measure
in H
sigma-algebra M
Borel
that Hn
en
-)
Wn
To prove
be
can
the rule:
by
1).
Mn.
sigma-algebra
wn(A)
this,
=
.3ince
=
=
sigma-algebra
a
because
by
let v,
that the sequence and
only
(n
vn
SO,
=
can
Wn
IVn}n=1,2,3.... lim
v(M)
is called
=
arbitrary real-valued
Then, the n
of
sequence
=
is
a
easily verify that
we
get
a
then,
contradiction
sigma-algebra containing measures
1,
M}
measures n
in H.
in
a
complete
1, 2,3,.... We recall
=
to the
in M if
measure v
I W(x)vn(dx)
W(x)v(dx) M
measure w
from Definition
measures
jWn} weakly
one can
as
IV.1.6 be
converges to the
countably
measure
w
additive. in H
as
oo.
Proof. First of c
vn(M)
Borel
bounded continuous functional 0 in M.
Lemma IV.1.10 Let the
--+
Borel
weakly converging
M an
as
recall
if
n-oo
for
H,
A E
with M. But
coinciding
the minimal
be considered
some
one can
1, 27 3,...) be nonnegative Borel
metric space M such that
separable
Then,
contains all open and closed subsets of
sigma-algebra is
a
Consider the set M,
n
in H contained in M and not
all open and closed sets.
Now,
Hn for
n
Mn7 M1 : Mn by the supposition.
definition the Borel
(we
there exists A E M such that
n
obviously M,
A E M
wn(A n Hn),
of all Borel subsets A of H such that A n Hn E M1. n
M1 is
onto the whole
it suffices to show that A n Hn is
Suppose the opposite. Then, A Clearly, M' fC C H : C
and M1n C
sigma-algebra
sequence of finite-
naturally extended
subset of Hn if A E M. A n Hn
our
jWn}n=1,2,3,.-
measures
show that each
the
on
we
> 0 there exists
a
all,
compact
set
prove
in the
proof
If, C H such that
of Theorem IV.1.8 that for any
w(If,)
> I
-
e
and
1,Vn(K,)
> 1
-
c
INVARIANT MEASURES
CLIAPTER 4.
116
for all
integer
n
Further, let
0.
>
take
us
arbitrary real-valued
an
continuous bounded
functional W defined in H and prove that lim n-oo
j W(x)w,,(dx)
W(x)w(dx).
Take also
arbitrary
an
0.
>
E
(IV.1.4)
H
H
Then,
easily verify that
one can
there exists 6
=
8(e)
>
0
such that
IWW for any
x
E
If, and
Let ifn
=
if,
satisfying I x
y E H n
H,,,
n
f
=
-
=
sup I W (x) 1,
so
(IV.1.5)
0
o(x)wn(dx) <eM,
Kn
H
where M
W(Y) I
-
that
we
have
XEH
liminf n-oo
I W(X)Wn(dx)
W(x)wn(dx) Kn
H
Let of
c
>
show the existence of C > 0,
us now
depending only
n-oo
I W(x)w(dx) I W(X)Wn(dX)
W and
independent
Clearly,
Y E H:
Then, K,
C
i.
Y
e.
=
Kn,
c
0, relations (IV.1-6) and (IV.1.7) together
>
the statement of the lemma. Let
Y1 + Y27 yi E
for all
Hn7
I
Y2
E
Hn, IY21H
sufficiently large numbers
n.
K.,,,,
0
real-valued
a
4D(X)Wn(dx)
where Q is
an
arbitrary
c
and
n.
continuous functional in H bounded
nonnegative
bounded subsets of H. Consider the
Y. (Q)
of
independent
quantities and
p(Q)
4D(x)w(dx),
bounded Borel subset of H.
Lemma IV.1.11 liminf Yn(Q)
p(Q) for
!
arbitrary
an
open bounded set Q C H.
n-oo
liM SUP [In (K) :5
p(K) for
an
arbitrary closed
bounded set K C H.
n-oo
Proof is standard. Here us
prove, for
Fix
an
example,
arbitrary
satisfying
the
3.
>
0 and take
a
second
(the
can
real-valued functional
be
0,(u)
paper
continuous in H and
following properties:
0,(u) 0,(u) 0 1 0, (u)
1. 0 < 2.
c
[109]. Let proved by analogy).
present its variant taken from the
we
the first statement
< I
for all
=
for any
=
if
Phen, according
u
u
E
X;
Q;
u
E Q and dist
to Lemma
IV.1.10,
lim inf Yn (Q) > lim n-oo
(u, aQ) we
>
c.
have:
0,(u)y.,,(du)
=
f 0, (u)
4D (u) w (du)
=
fV),(u)dy(u). 0
Taking here the limit over a sequence
0,ju) satisfying
0e(U)(D(U)Wn(dU)
n-oo
Q
the above
lim
=
'b_00
c,
properties
-*
+0 and
with
c
=
an
E,
arbitrary sequence of functionals
we
get liminf tln( I) ! P(Q)-O n-oo
118
CHAPTER 4.
An invariant
4.2 In this
section,
for the NLSE
measure
shall construct
we
invariant
an
INVARIANT MEASURES
measure
for the
NLSE;
we
follow the
paper [107]. Let A > 0, the space L2(0, A) be real and let X be the direct product L2 (0, A) L2 (0, A) of two samples of the space L2 (0, A) equipped with the scalar
product
(7-VI W2) X
:--
i
where
(Ul U2) L2 (0,A)
(ui,vi), ui,viEL2(0,A),
wi
+
I
i
=:
(V1 V2) L2 (0,A) i
1,2, and the corresponding
norm
jjujjx
'I
(u, U)2X. as a
Let
us
consider the
following problem
for the NLSE written in the real form
system of equations for real and imaginary parts of the unknown function:
*1+ U2 t X
*2t
_
UIX
+
f(X, (UI)2
+
(U2)2)U2
=
0,
xE(O,A), tER,
(IV.2.1)
_
f(X, (Ul)2
+
(U2)2)Ul
=
0,
xE(O,A), tER,
(IV.2.2)
X
u'(0, t)
=
u'(A, t)
u'(x, to) Formally
the
=
problem (IV.2.1)-(IV.2.4)
u'o
is
=
0,
t E
R,
(IV.2.3)
L2 (0, A).
E
equivalent
(IV.2.4)
to the
integral equation
t
u(t)
A(t
=
-
to)uo
+
j B(t
-
s)[f(.' JU(s)j2)u(s)]ds,
(IV.2-5)
to
where
(to
be
(Ul (t), U2(t)) interpreted as X),
u(t)
=
cos(tD) sin(tD)
A(t)
-
In this
section,
(f)
Let
possessing
a
our
is the unknown function with values in Uo
=
sin(tD) cos(tD)
hypothesis
f (x, s) be
a
continuous
on
derivative
all
(x,.5)
E
[0, A]
x
+
sin(tD) cos(tD)
B(t)
the function
if(x, 5) 1 for
and
f
real-valued continuous
partial
a
functional space
(Ul,U2), 0 0
is the
following.
function of (x, s)
f,,(x, s)
If.,'(x, s) I
cos(tD) sin(tD)
-
E
[0, A]
and let there exist C
x
[0, +oo)
0 such that
C
[0, +c*).
One may define X-solutions of the
problem (IV.2.1)-(IV.2.4) by analogy
H'-solutions of the NLSE from Definition 1.2.2.
with
AN INVARIANT MEASURE FOR THE NLSE
4.2.
Proposition IV.2.1 Under -2
119
hypothesis (f) an arbitrary solution u(-, t) E (0, A)) of the problem (IV. 2. 1)- (IV. 2.4) satisfies equathe
-2
C(I; X)nC'(I; H (0, A) x H tion (IV.2.5), and conversely, any solution u(-,t) E C(I;X) of equation (IV.2.5) -2 a solution of the problem (IV. 2. 1)- (IV. 2.4) of the class C(I; X) n C'(I; H (0, A) -2 H (0, A)) (here I [to T1, to + T2] where T1, T2 > 0). =
Proof
Now
can
-
be made
hypothesis (f)
(a) for (IV.2.1)-(IV.2.4);
a
unique global X-solution u(., t) of the problem
t) be the function transforming any pair (u'0 U20 ) E X and t E R h((ul, 0 U2), 0 X -solution (ul (-, t + to), u2(., t + to)) of the problem (IV.2.1)-(IV.2.4) taken let
,
at the moment
phase
1.2.313
present the main result of this section.
we can
any uo E X there exists
into the
proof of Proposition
with the
by analogy
Theorem IV.2.2 Under the
(b)
is x
space
of
time t + to.
Then,
the
h is
function
a
dynamical system with
the
X;
d
(C) dt I JU(., t) 112X (IV-2-4); (d) let w be the D`
0
0
D-1
)
0
=
for
arbitrary
an
centered Gaussian
with the correlation
measure
Since S is
in the space X.
of the problem (IV.2.1)-
above X-solution
operator of
an
trace
operator S
class,
the
measure
A
countably additive.
is
w
Let also
f F(x, (UI)2
(D(u',u2)
+
(U2)2 )dx
where
F(s)
0
2
f f (x, p)dp (the functional 4
is
obviously
real-valued and continuous in the space X
0
bounded
on
bounded subsets
of X). Then,
Q is
measure
an
for
arbitrary
the
Borel subset
dynamical
of X)
well-defined
is
system h with the
phase
Proof of Theorem JV.2.2. The map from the
C(I; X)
forms the space
measure
in X
e(1(U1'U2) w(dul dU2)
Y(Q)
(here
the Borel
into itself and is
a
in X and it is
an
invariant
space X.
right-hand
contraction for
side of
(IV.2.5)
sufficiently
trans-
small val-
T,to + T]). Therefore, depending only on JJuoJJx (here [to equation (IV.2.5) has a unique local solution of the class C(I;X). Further, let ues
of T
I
> 0
IA,,, en}n=0,1,2.... tor D and let
L2(0,A)
be
Xn
onto the
=
eigenvalues
and the
span(eo,..., en)
and
=
-
corresponding eigenfunctions
of the opera-
Pn be the orthogonal projector in the
subspace Xn. Let also Xn
=
Xn & Xn.
Consider the
space
following
120
CHAPTER 4.
INVARIANT MEASURES
problem approximating the problem (IV.2.1)-(IV.2.4): 1
2
Un 2
1
Un
_U
t
n ,X
U n
Let pn
Pn
0
0
Pn
(Ul)2
+
(U2)2)U2]
0,
t E
R,
(IV.2.6)
X-Pn[f(X, (Ul)2
+
(U2)2)Ul]
0,
t E
R,
(IV.2.7)
+ Un,xx + Pn[f(X,
t
)
n
(X, to)
be the
in the space X. Let also gi
the
eigenelements
PnUI(X), 0
=
n
n
n
n
U2(X,to)=pnU2(X). 0
orthogonal projector
=
onto the
=
-
of the operator S.
(IV.2.8)
n
(eO, 0), 92 (0, eO), f9n}n=1,2,3,... is an orthonormal
the system
Then,
n
Clearly,
-
-
7
=
92n+1
subspace
Xn
(en) 0), 92n+2
basis in the space X
for any
positive integer
Xn (D Xn =
(0, en)
7
consisting
n, the
....
of
problem
(IV.2.6)-(IV.2.8) has a unique local solution un(x, t) (U1(X't)'U2(X' t)) E C (I; Xn) (as it is well known, in a finite-dimensional linear space any two norms are equivalent, =
n
and
we
that the space Xn is
mean
addition,
equipped
for any
Therefore,
(IV.2.6)-(IV.2.8)
n
has
=
a
1, 2, 3,
unique global
(IV.2.8) satisfy
the
equations (n
=
=
of the space
X).
In
0 for these solutions.
uo
solution
it is clear that the above
Further,
norm
X the problem (Ul0 (.), U20 Un(*) t) C(R; Xn). solutions Un(*,t) of the problem (IV.2.6)-
and for any
...
with the
-dt-JlUn(*, t) 112X
the direct verification shows that
n
E
1, 2, 3,
t
Un(', t)
=
A(t
_
to)pnUo
f B(t
+
-
S)pn V(.' JUn(*, S)12 )Un(*) s)]ds.
(IV.2.9)
to
Hence,
u(., t)
from
(IV.2.5)
and
(IV.2.9)
has for those values of t for which the solution
one
exists: t
JU(', t)
-
Un
(', t) I IX
:5
C1JJUo
_
pnUoJJX
+
C2
I I JUn(*, S)
-
u(., s) I lxds+
to t
+C3
J I JU(., 3)
_
pnU(., s) I Jxds.
(IV.2:10)
to
Here the constants the solution
right-hand respect
u(-, t)
C1, C2, C3 do exist for t E
side of this
to t E
depend
not
[to, to
+
T]
on
where T
inequality obviously tends
[to, to + TI,
therefore
we
the initial value uo,
to
> 0.
Then,
zero as n -4
get from inequality
to and t. Let
the third term in the +00
uniformly
(IV.2.10) by the
lemma that
By analogy,
lim
max
n-oo
tE[to,to+Tl
if the solution
u(-, t)
JJU(',t)-Un(',t)JJX=O-
exists
lim
max
n-oo
tE[to-T,to]
on a
segment [to
-
T, to],
JJU(',t)-Un(',t)JJX=O-
T >
with
Gronwell's
0, then
AN INVARIANT MEASURE FOR THE NLSE
4.2.
121
Hence,
for all segments I fact
implies,
in
of
solvability
max
tEI
T1, to
-
I Ju(-, t)
particular,
it is easy to
for any
(IV.2.11)
0
=
of Theorem IV.2.2
(c)
uo E
if
verify that,
u,,(., t) I Ix
-
of the existence of the solution
T2]
+
the statement
equation (IV.2-5)
Further,
(IV-2.5),
[to
=
lim n-oo
This
the
global
and, hence,
X.
function
a
u(-, t).
then for any fixed t E R this function is
u(., t)
G
C(R; X)
solution of the
a
satisfies
equation
following equation:
T
u(.,,r)
A(T
=
+IB(,r
t)u(-, t)
-
_
S) [f(.' I U (.' 5) 12) U (.' s)]ds,
r
E
R,
t
which,
for any fixed t has
earlier,
as
any fixed t the map uo
the transformation
(t
>
uo
u(., t) is u(., t) as a
--+
--+
X-solution.
unique global
a
one-to-one from X into X.
for
Therefore,
The
continuity
of
map from X into X follows from the estimate
to) t
U
('i t)
-
Vt)
X
! C1
U
('; to)
-
(*) to)
V
X
+
C2
)rI lu(.,,.s)
-
v(-, s) I Jxds,
to
where
and
u(., t)
estimate for t
0
>
I JUn(i t)
max
of
and
equation (IV.2.5), and
a
similar
(c) of Theorem IV.2.2 are proved.
0 there exists 8 > 0 such that
-
Vn(*l t) I IX
no.
y (Q)
a
number
Further,
:5 p (B) +
we
no > 0 such that dist (hn (u,
(B)
+
,
get by Lemmas IV.1.11 and IV.2.4
< lim inf yn
c
t); X21)
c'
by open
Generally, it may
we
sets
obtain the equality
containing
the invariant
happen
h(Q, t))
that
A.
I.L(X)
y(h(Q, t)).
IL(A) jz(h(A, t)) by the apThus, Theorem IV.2.2 is proved. 0
measure =
=
p
=
given by Theorem IV.2.2
+oo.
However, according
can
to the
CHAPTER 4.
124
(c)
statement
of this
h.
dynamical system space of the a
ball,
physical
an
invariant set of the
choose any of these balls for
a new
on
phase
any of such
jt(BR(0)) < +oo for any R > 0. By analogy, in view of jL(B,(a)) < +oo for any a c X and r > 0. As a corollary, all points of the space X (in the sense of the measure w) are
0 is
system h. Since the functional 4D is bounded
Remark IV.2.7 The two
we
BR(O)
0
0 in
the
case.
An infinite series of invariant
4.3
measures
for
the KdVE In this
the
section,
consider the
we
following Cauchy problem periodic
variable for the standard KdVE with
spatial
uo(x
+
A)
uo(X),
=
We shall construct tion laws
E,
n
>
an
is the
norm
(IV. 3. 1)
u(x
X, t E
R,
(IV.3.2)
+
A, t)
E
R,
=
u(x, t),
(X, to)
and A
Uo
=
(X),
(IV-3-3)
is fixed. Here
> 0
infinite series of invariant
we
follow the paper
associated with
measures
[111].
conserva-
3, given by Theorem 1.1.5 for dynamical systems generated by the suitable
on
in the space
phase
Throughout
spaces.
n
Hper(A) by
for
simplicity.
this section
we
shall
Our first related result
following. Theorem IV.3.1 Let
Then,
the
function
the rule: hn (Uo'
hn
Hpn,r(A).
The
=
>
2, A
>
Hpner(A)
into
Hpn ,JA) defined for any uo E Hpner(A) by is the Hpner(A) -solution of the problem
integer
from
t) U(., t (IV.3.1)-(1V.3.3) given by
this
u:
x,tER,
x
problem (IV.3.1)-(IV.3.3) denote the
==
ut+uux+uxxx=O,
U
where
f (u)
with respect to
+
to),
n
where
Theorem
functionals Eo,
...'
0, T
>
0,
t E R and to E R be
arbitrary.
u(., t)
L1.5,
is
a
dynamical system
E,,,-, from Theorem L 1. 5
are
with the
phase
space
conservation laws
for
dynamical system. Let
n
> 2
be
integer
the correlation operator S
and Wn be the centered Gaussian =
(I + Dn-1)(Dn
+
I)-1.
Since
measure
clearly S
in
Hpn,-r'(A)
is
an
with
operator of
4.3.
AN INFINITE SERIES OF INVARIANT MEASURES FOR THE KDVE 125
trace
class in
Hp',,-,'(A), the measure w' is countably additive.
Let also for
u
Hpn,,,(A)
E
A
1
Jn (u)
En (u)
=
-
2
1 f[U(n)]2 x
U2 }dx
+
I
En(u)
=
-
2
(S_1U) U)HP1_(A)
=
0
A
JJCnU [U(n-1)]2
2 u
x
_
2
qn(Ui
U
(n-2))Idx.
x
0
For
arbitrary
an
Borel Q C
Hpne-r'(A)
set
we
e-Jn(u)dWn(U)'
,p(Q)
where Q C
Hpn,-'(A)
is
a
.r
Borel set. The main result of this section is the
Theorem IV.3.2 For any and it is
measure
an
IV. 3. 1. For any
orem
Rd
integer
invariant
sufficiently large U
Hpn,-' (A)
G
:
Since due to Theorem IV.3.1 Rd is it
can
be taken
for
phase
a new
applicable. So,
almost all
Wn )
according
are
Poisson
stable
form
a
latter).
Now
we
for
0
0 < ,n (Rd) < + 00, wh e re
>
invariant set
an
therefore space
and,
in
1, 2,3,.... Theii., of
=
Ao
3
n
measure
following.
orthogonal projector
be the
orthogonal subspace LIn. Consider
in
proj ector the
following problem: u' + Pm [U'U'l + Um t x xx x
=
0,
x
E
(0, A),
t E
R,
(W-3.4)
CHAPTER 4.
126
U, (X,
Clearly, for
C'Qto of
-
T, to + T]; L,,,) for Since it
Hpn,,(A)).
it has
HP",(A)
E
any uo
some
be
can
to)
easily
is
and, hence,
(the topology
global (it
can
a
Proposition IV.3.3 be such that
0,
=
we
E
norm
have:
1Um(*,tO)jL2(0,A)
=
problem (IV.3.4),(IV-3.5)
for any
uo E
Hn Pe
r(A)
R). (We apply norms are equivalent).
here the
finite-dimensional linear space any two
this solution is
addition,
the
generated by
-dt -Ej(um)
=
be continued onto the entire real line t E
known fact that in In
in L,,, is
-dt -Eo(u')
verified that
the solution of the
(IV-3-5)
unique classical local solution u'(x, t)
1UM(*7t)1L2(0,A) for all t,
P"' uo (x).
=
a
T > 0
INVARIANT MEASURES
differentiable in
obviously infinitely
Let
> 2 be
n
k
integer, "
uo E
Hpnr(A)
and uO
and t.
x
and let
k
a
sequence
u
k
0
Hpn,,,(A) oo strongly in u(., t) as k Hpn,,r(A), where u(.,t) is the Hpn,,r(A) -solution ofthe problem (IV.3.1)-(1V.3.3) and umk(-,t) is the Lmk -solution of equation (IV. 3.4) with m mk and with the initial data as
k
--*
mk
--
Then
oo.
+oo
as
-+
any t E R
for
umk (.,
Lmh
E
oo, uo
t)
---
--+
strongly
uo
in
--+
=
um'; (.,to) Before
proving this
statement consider
Lemma IV.3.4 For any
R(n, d)
0, such that
>
if u
E
1julln
2
1 E,, (um (-, t))
-
tEI 2
n
I IDxnum(., t)122( L 0,A)
-
IDx Um (.' to) IL2(0,A) I
-
(t
>
to):
En (um (-, to))
On(IIUM(*)t)lln-1)
On(IIUM('7tO)IIn-1),
-
where
an(s) and On(s) are functions continuous and nondecreasing on the half-line [0, + oo) (n > 2). Indeed, fix an arbitrary R > 0 and let I I UO I I < R. Then t aking into account that as earlier I I um (-, t) Ill :5 C(R) for all t, we get step by step: n
IIUM(*7t)112 and Lemma IV.3.6 is
:5
C,,,(R),
proved. 0
Lemma IV.3.7 For any takes
C2(R),..., IIUM(*it)lln
0
By analogy
-L
k1iM PMk (UMk (.' t)Umk (.' t))
0
X
oo
for any t, and Lemma IV.3.8 is
Now
just
as
in the
proof
proved.El
of Theorem
IUMk (.' t)
-
1.1.5,
one can
U(., t) I In
__
0
prove that for any t
=
0
CHAPTER 4.
130
k
as
--+
if uo C-
oo
IV.3.9 Let
Corollary lim
[E" (um (-, t
and
Hpn,,,(A),
-
n
Proposition IV.3.3 be integer.
> 3
E" (um (., to]
==
is
INVARIANT MEASURES
proved.0
Then, for
Hpn,,-,'(A)
any uo E
and any t
0.
M-00
Proof repeats the
of Lemma IV.3.8 in view of the
proof
proved Proposition
IV.3.3. D
to prove Theorem
Below,
Proposition IV.3.10 8
IV.3.2,
For any
n
>
shall also need three statements.
2,
uo E
Hp'e,(A),
e
>
0 and t E R there exists
such that
> 0
I 1Um(-' 0 for
we
m
any
1,2,3,... and
=
an
UN., 0 11.
-
3
function n(s) nondecreasing and
I En (Um (') t)) for
all
m
=
1, 2,3,... and
uo E
-
a
contradiction.
be
integer.
continuous
En (Um (*
i
0
on
to)) 1
-+
uo
0
==
as
ul). Then, k
--+
there
oo, such that
> 6,
__+
7
in
with
Pm, uo 11,,
-
with uo
accept that the sequence ml, is unbounded
But then
>
Jjumj(-,to)-um(-,to)jj,
0 there exist
r
E
(0, ro)
0 such that
En (Um (. t)) i
-
En (um (,) to)) I
mo
m
Fix
I luo
if
-
arbitrary
an
f-
V1 1,,_1
a B, (ul) B, (ul), possessing the above property. by these balls and let ml,..., mi be numbers such that for
Let
takes uo E
place K if
if
m
-
-,
finite
and
set K
covering
Bjuj). Then, obviously (IV.3.6) Proposition IV.3.12 is proved.0
a
ball
of the set K
any i the relation
and uo E
> rni
> maxmi,
m
.
be
(IV.3.6)
is also valid for all
i
Fix
Hpn,-,'(A) e.
arbitrary integer
an
consisting
of
n
From here
> 3.
eigenvectors
the above-defined functions with the
el, are
Let also wi
(I
=
A' )-'(l
+
A '-'),
+
is the orthonormal basis in
jek}
of the operator S of the above-indicated kind
where i
change
of
n
by
n
0, 1, 2,.... Then,
=
-
(i. definition).
I in the
wi are
eigenvalues
of
S. Consider in the
subspaces Lk
the finite- dimensional Gaussian
Hpn,-'(A)
C
spanned
defined
measures Wk
over
jei}i=o,1,2'...'2k
vectors
the rule
by
2k
2k I
2A;+1I I Wi
-
Wk(Q)
=
(27r)
-
7
i=O
where Q
Then, the
u
W1, is
a
measures
E
Lk I [(Ui eO)n-1)
Borel Wk
IV.1.10 the sequence sian
measure
Wn
.
For
7
Borel
as
obviously
...
Fj
E
and F C R 2k+1
According
are
to results from
in
measures
Hpn,,(A)
HPn,;1(A),
Borel sets.
Section 4.1,
and
by
Lemma
we
also set
e_j'(u)dWk(U)-
Ak (f2)
Since the functional J,, is
Idzo &2ki
converges to the infinite-dimensional Gaus-
fWk}k=1,2.... weakly Borel set Q C
Z? E W` i _0
e
(U, e2k)n-1]
be considered
a
f
2
F
in Lk for any k.
measure
can
...
1
continuous in
subsets of this space, the
Hpne-,' (A)
and bounded
on
bounded
Let measures ltk and /,n are well-defined in HPn,,,1(A). h, (u, t) be the dynamical system with the phase space L,,, generated by the system
(IV.3.4), (IV. 3.5)
_
that the function h,,, (.,
t) for any fixed t transforms L.. into L.. and any uo E L maps into the solution u- (-, t + to) of the problem (IV. 3.4), (IV.3.5) taken at tl, e moment of time t + to. Obviously h,,, (., t) for a fixed t transforms also Hpn,-,'(A) into I, according to the rule h,,, (u, t) h,,, (P.,, u, t). so
=
Lemma IV.3.13 Let t E R and Q C lim
W2'-'
(p,,, (h, (Q, t))
-
be
a
closed bounded set.
it,,, (Q))
=
Then
0.
M-00
Proof. Let
(ZO (t)
7
Z2m
(t))
us
rewrite the system
where u- (x,
t)
=
zo
i(t)
(IV.3.4),(IV.3.5) in the coordinates z(t) (t) (x) + + Z2m (t) e2m (X). Then, we get eo
=
-
-
-
JV-,H(z(t)),
(IV.3.7)
AN INFINITE SERIES OF INVARIANT MEASURES FOR THE KDVE 133
4.3.
(to)
z
(uo, ei),,-,,
=
i
H(z) Ej(zoeo + + -2 2me2m) and J is _2irk matrix (i. e. J* -A (J)2k-1,2k A 1,2,...,m)and (J)k,l 0 for all other values where
::::::
---
=
Let
Theorem
det( )
(1 (jrk)2n-2) +
measure
=:
A
of the indexes
k,1
1 for all t.
azo,3
IV.1.3, the Lebesgue
(2m + 1) -(J)2k,2k-I (k
skew-symmetric (2m + 1)
a
t 2au
prove that
us
(IV.3.8)
2m,
0,
=
x
0,1,...,2m.
=
Indeed, according
to
ij=5_,_2m
f dzo
orm(Q)
...
dZ2,,, is
invariant
an
mea-
n sure
for the
with the
dynamical system
phase
L,, generated by the problem
space
(IV.3.7),(IV.3.8). Therefore, orm
(hm (Q, t))
I
=
dzo
...
Vdzo dZ2,rn
dZ2m
dzo
...
...
dZ2m
h- (O,t)
for
arbitrary Borel
an
this
set 9 C R 2m+1
that
immediately implies Let
us
arguments,
take
V
=-:
continuity of the function
17,
1.
arbitrary closed
an
In view of the
bounded set Q C
In view of the above
Hp'e-r'(A).
get:
we
ym (hn (9,
eEn (Pmu) -E,, (hm (u,t)) dyn (u).
t))
Further, ym (Q)
-
therefore, according
integrand respect K C
proof
to
in the
m
jn(P u)-En(hm(u,t)) I dym (u),
t))
Proposition IV.3.11 and
to
right-hand
integer
Hpne-r'(A)
pm (hm (Q,
side of this
0 and
>
such that
E Q.
u
p(Q \ K)
of Theorem IV.1.8.
equality
< c,
Take
Lemma
is
an
a
IV.3.6,
we
obtain that the
function bounded
arbitrary
c
>
the existence of which
0 and
can
be
uniformly with a
compact
proved
as
set
in the
By Proposition IV.3.12,
[ttm (K
lim
n
Q)
-
tt,, (h,,, (K n
Q, t))]
=
0,
M-00
hence, by Proposition IV.3.11,
we
get the relation
lim sup [ftm (Q)
-
ym (hm (Q,
t))]
0, yields the
statement of Lemma IV.3.13.0
IV.3.14 For any bounded open set 0 c lim M-00
I tt,, (Q)
-
p,,, (h
(Q, t)) I
Hpn,-'(A) =
0.
and
for
any t E R
134
CHAPTER 4. Lemma IV.3.15 Let Q C
,,n(Q)
,n (h
=
Proof.
n-1
Take
too.
dist(A, B)
v
E
Let K,
=
=:
u
is
bounded open
a
exists
compact
a
set
hn-1 (K, t).
=
By Proposition IV.3.10 for
Hpn,,-r'(A)
Then, there
0.
>
c
-
uEA, vEB
0.
Then
c. \ K) Then, If, is a compact t) f2j. Let a minf dist(K, ffl); dist(KI, aQ,)}, where JJU Vjjn-j and aA is a boundary of a set A C Hpne-,'(A).
inf
>
arbitrary
0
of 0
nonzero
and
I_L(B) < 00 -C(l+s di.) :
0
0 and
case
and obtain sufficient
y similar to those from Theorem IV.2.2 to be
finite for any ball B C X. The obtained conditions
(as-
proved. However,
cubic nonlinear
measures
besides
measures
nonlinearities such
[107],
is not
result of the paper
abstract construction from
measure
case
nonlinearity. Methods exploited
related to invariant
our
measures
of the KdVE and E in the
[4,11,18,19,24,30,55,65,67,86,107-
is constructed for
and
[11],
In
superlinear
constants. In
in
NLSE. Similar
a
important details of the proof
in this paper.
carefully reestablishes
example,
easily follows
invariant
Unfortunately,
tion.
for
the invariance of these
109,111-1131. equation.
properties
number of papers devoted to their
nonlinear
is constructed for
measure
considered,
are
the invariance in
theorem
investigations
sociated with the energy conservation law El in the case
recurrence
the KdVE and NLSE which
indicated earlier. One of the first results in this direction is obtained
are
where
explain the
to
recurrence
a
dynamical systems generated by
struction for
in
of the Poincar6
application
Here
[16,20,53,66]).
approaches
are
dynamical systems generated by
of
trajectories
not based
note that there
we
bounded
on anY
proved. El.
completely
Thus, Theorem IV.3.2 is
(h n-1 (A, t))
IV.3.4, the continuity of
and from their boundedness
Hpn,-'(A)
n =
open sets from outside. The
by
last two statements of Theorem IV.3.2 follow from Lemma the functionals
,n (A)
d, > 0, d2 E (0, 1) AJuJP this implies: p > 0 if A < 0 and f(X, S) :5 C(l + S d2) for all x, s. For f(JU12) this in A for 0. > Unfortunately, paper an important question remains open p E (0, 2) about the well-posedness of the initial-boundary value problem (IV.2.1)-(IV.2.4) with for any ball B C X if there exist C
=
N
=
1 in this
superlinear
result is obtained
this
problem
for
by an
J.
case
with initial data from
a
space like
L2_ The required
Bourgain [16,17] who proved the well-posedness
arbitrary
in
a sense
of
A. This allowed the author of this paper to construct
an
invariant
measure
for the one-dimensional NLSE with the power
nonlinearity and
(0, 5) (see [18]). [65].
A result in this
to show its boundedness in the above
direction for the cubic NLSE is also
[67],
In the paper
law
sense
for p E
presented
in the paper
invariant measure, associated with
an
IV.3.2
the existence of
on
periodic
in the
a
spatial
may be
posed:
Eo, El, E2 Eo. In this
comments
a
for the KdVE case,
our
for the
measures
problem
=
0,
(or
the
NLSE)?
question
space Hn-' corresponds
phase
of the
For
example,
consider the
However,
we can
main difficulties in the way of
an
to the nth conservation law.
should prove
a
required
measure,
evolution
problem
for the KdVE with initial
to construct
corresponding
H-1
invariant
constructing
we
at least from
(or
do not know any results like that. In
we
associated with the lowest
is not answered yet.
space similar to the Sobolev space
Unfortunately,
measures
it. One could observe that in Theorems IV.2.2 and IV.3.2
hypothesize that
we can
well-posedness
data from
on
the
measure on
Therefore,
analogous
\IU12U
Ux +
there invariant
are
conservation law some
con-
variable for the usual cubic NLSE
conservation laws
invariant
is
function,
constant, is presented in the paper [112]. In this connection, the follow-
ing question
make
conservation
higher
to the result of Theorem
infinite sequence of invariant
an
iUt + where \ is
a
the square of the second derivative of the unknown
containing
structed for the sinh-Gordon equation. A result
the
INVARIANT MEASURES
C11APTER 4.
136
H-2-', 6 > 0). opinion, this is one of the
our
measures
corresponding
to the
above conservation law. in the paper
Finally,
[113],
the
following Cauchy problem
for the NLSE written
in the real form '
I
U
-
t
UX X + V (X) U, +
U2+ U'X t
-
X
V(X)U'
(U')'
+
(U2)2)U2
=
0,
x,t
E
R,
(IV.4.1)
f(X, (U')'
+
(U2)2)Ul
=
0,
X,t
E
R,
(IV.4.2)
f (X,
-
u'(x, to) where
V(x)
is
is assumed that the function The main
Theorem IV.2.2.
eigenvalues
to +oo
of the operator
v,,
--
E
x
(IV.4-3)
1, 2,
=
R,
is considered.
In this paper, it
satisfies conditions similar to those introduced in
hypothesis
IxI
as
f
i
u', 0
real-valued function of
a
positive, tends
=
oo
(_
d2 d 2
on
the
potential V
and increases +
V(X)
as
IxI
Yi satisfy
*
---
is that this function is oo
so
rapidly
the condition
E v,,,
that the
