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
(generalized)
1.1
The
1.2
The nonlinear
1.3
On the
1.4
Additional
Chapter
2.
equations.
Results
on
existence
Vries equation Korteweg-de equation Schr6dinger (NLSE)
blowing
36 37
problems
Stationary
39
Existence
of solutions.
An ODE approach
Existence
of solutions.
A variational
2.3
The concentration-
2.4
On basis
2.5
Additional
3.3 3.4
Additional
3.2
Chapter
4.
of P.L.
49
Lions
56
of solutions
62
remarks
76
of solutions
79
of soliton-like
of kinks
for
of solutions
Invariant
42
method
method
compactness of systems
properties
Stability
Stability Stability Stability
3.1
10 26
up of solutions
2.2
3.
(KdVE)
remarks.
2.1
Chapter
9
solutions
the
80
KdVE
of the
90
NLSE
nonvanishing
as
jxj
remarks
oo
94 103
105
measures
4.1
On Gaussian
measures
4.2
An invariant
measure
4.3
An infinite
series
4.4
Additional
remarks
in Hilbert
for
the
of invariant
107
spaces
NLSE measures
118
for
the
KdVE
124
135
Bibliography
137
Index
147
Introduction
that
field
leading
are
approach properties makes more
possible
it
general
study).
the
present
In
qualitative
results
studies
dealt
with
on
of
travelling
investigate book,
author
to
the
problems
waves)
dynamical
stability
systems
twenty
substituted
solitary
of
in
by
following
(generalized)
main
material
is
These
topics.
these
equations,
(for example, consideration,
under
of invariant
construction
Vries
for
kinds
special
equations
Korteweg-de
generated
the
four
are
of
the
and the
waves,
of the
problems
initial-value
for
when solutions
arising are
There
a
evolutionary,
and
stationary
So, the selection
years.
interests.
of solutions
existence
both
in
of the
methods
and
problems
approach
(maybe
problems
of
class
some
surveys
scientific
author's
the
standing
of the
about
during
wider
blowing-up,
or
and this
etc.,
is
prob-
of various
stability
as
there
The latter
equations.
equations,
these
consideration,
under
equations
such
of solutions
hand,
other
well-posedness
the
subject,
this
known nonlinear
the
on
of differential on
by generated an essentially
the
and,
narrow
(on
problem
that
equations
these
currently
of
class
theory
behavior
systems
stationary or
problems for
of
related
mainly are
theory
he has
that
to
sufficiently
investigations
the
dynamical
of
the
time,
the
is
methods
by
problem;
scattering
inverse
qualitative
the
equations,
these
same
is
includes
particular
in
for
lems
called
approach,
another
At the
method
by this
PDEs solvable
of the
method
the
[89,94]).
example
for
see,
by
solvable
called
from
mono-
discoveries
field
this
problems
in the
example,
for
scattering
inverse
physicists
a
in view
Physical
mathematical
related
equations
nonlinear certain of studying possibility the to were quantum analyze developed
has grown into
-
and
observed,
are
One of the
partial
of nonlinear
kind
problems.
of the
novelty
consideration
[60].
special
a
mathematicians
of both
and of the
Makhankov
of
theory
the
-
solutions
attention
under
equations
V.G.
by
the
applications
the
to
graph
(PDEs) possessing
attracts
important
of its
of solitons
theory
the
30 years
equations
differential
large
last
the
During
measures
Schr6dinger
and nonlinear
equations. the
We consider
Ut
and the
Schr6dinger
nonlinear
+
Korteweg-de
f (U)U.,
+ UXXX
i is the
imaginary
and
complex
the
NLSE with
in the
unit,
second),
u
u(x, t)
=
t E
R,
x
0
(NLSE)
equation
iut + Au + f where
=
(KdVE)
equation
Vries
is
(Jul')u an
0,
unknown function the
E R in
=
case
of the
(real
in the
KdVE and
first
x
E
A
=
case
R' for N
a
positive
integer
N, f (-)
is
a
smooth
real
function
and
E k=1
P.E. Zhidkov: LNM 1756, pp. 1 - 4, 2001 © Springer-Verlag Berlin Heidelberg 2001
82 aX2
k
2
Laplacian.
is the
As 2) respectively,
Chapter
u
qonsider
the
x) (it
equation
what
(as JxJ
oo
NLSE,
for
following A0
which
we
0
if
the
in
NLSE,
In what
is
supplement
with
nonlinear
Loo
_
elliptic
boundary
some
problem
A similar
problem
0(k)(00)
type
Difficulties
=
nontrivial
solutions
the
kinds).
In this
integer
any
argument
r
I > 0 there
exists
Ix I,
method
are
the
and
of
in finite
intervals
Chapter Lyapunov set
of the
qualitative
method.
results
sense.
X, equipped
to
3 is devoted
to the
Omitting a
those
some
distance
of
of P.L.
Lions.
being
differential
indicated
stability
2).
Chapter
of
functions
argument
is
the
as
a
f r
=
following:
of
W. for
function
of the
waves.
These
addition,
briefly
(ODEs)
in
consider
this
(for example,
the
chapter
we
L2)
in
Sturm-Liouville-type
one-dimensional
1,
==
of the
equations
we
basis
a
for
solitary
of
latter, In
N
0.
>
r
us
which,
existence
of the
of
property
the
non-uniqueness the
on
for
ordinary
example
of nonlinear
similar
with
the
half-line the
(see
solved
only
problem
our
the
into
order:
conditions
(for example,
f interesting of
00
NLSE with
speaking,
exist
proving
an
easily
depending
theory As
on
on
of
method
eigenfunctions
systems
solution
the
satisfying
waves
generally
functions
waves
second
will
--+
x
example,
for
KdVE and
solitary
case,
standing
kinds as
le,
Ej
X
sufficiently
solutions
I roots
methods
compactness recent
upon
a
exactly
the
solutions for
two
variational
concentrationtouch
has
this
In
when such
result
=
be
can
consider
us
typical
We consider the
0, 1, 2)
=
occurs
Let
the
case,
(k
0
of
uniqueness
of these
of the
into
if necessary,
0.
=
KdVE. For
the
and
when N > 2.
arise
above
for
arises
of existence
for
waves
function,
limits
possessing
conditions,
061--
the
0,
real
a
expression
standing
specifying,
equation
=
is
solutions
expression
f(1012)0
+
the
function
the
0
notation,
and
Chapter
In
the
KdVE and
c R and
follows,
bounded
a
w
this
just
substitute
we
of the
case
where
NLSE). Substituting
the
and
problem
Cauchy
of the
when
arises
introduce
with).
waves
the
f (s)
functions
constants).
positive
are
v
well-posedness It
wt)
-
to
dealt
solitary
obtain
we
x
of the
case
being
of the
for the KdVE and the NLSE used further.
O(w,
=
the
on
and
a
problem.
is convenient is
the -+
u
in the
equation
be called
results
stationary
waves
e `O(w,
=
the
(where
problems
value
travelling
for
e-a.,2
+S21
1 contains
initial-boundary we
2
as
1
physics,
for
important
following:
the
are
Isl"
2,
examples,
Typical
for
problems
above. of
details,
this
R(., .),
there
solitary means
exists
waves,
that, a
which is understood if for
unique
an
arbitrary
solution
u(t),
in the
uo
from
t >
0,
a
of
3
the
under
equation
T(t),
belonging
R, if for
that
by
obtained
equation
(in
the
us
introduce
Let
of the
functions
(in particular)
call
we
distance
special
a
argument
by
x,
functions
two
some
p becomes
stability
of
a
solitary
family
two
t
0 in the
=
metric
W2.
time,
they
can
usual
the
stability
of
many authors
For
taken
and
solitary in the
wit)
-
we
shall
with
s
=
of
following
if
they
H'
is
remark
the
here
that
family
two-parameter ob w
now
> 0
arbitrary
(x, t) and
=
b
are
close
at
t
all
for
the
=
approach
the
at
Sobolev
or
point spaces,
velocities
0 in the
wi
sense
of the
same
sense.
solitary
waves
developed
was
of
stability
this
He called
p.
his
(a, b).
other
t > 0 in the
stability
distance
possesses
Therefore
E
of
the
because
usually
non-equal
have
close
proved Later,
Lebesgue
as
are
to the
their
NLSE,
such
w
each
to
the
with
first,
p;
KdVE
r)
-
investigate
to
distance
the
close
to be close
has
wave.
consider
of the
waves
of
v(x
=_
functions
parameter
L02t),
> 0
respect
solitary
a
-
waves
[7]
paper
the
on X
t
verified
easily
to this
by
results.
distance
the
p should
be modified.
should
It
form:
(u,vEH')
T"Y
family,
0
X
u(x)
condition
it is natural
d(u,v)=infllu(.)-e"yv(.--r)IIHI
where
0' (w, x) 0
if
H1 consisting
space
equivalent
of
second,
x;
spaces
all
solitary
if two
f (s)
O(W2,
for
sense
the
respect
depending and
Sobolev
real
reasons,
in
functional
be
form
the
For several
KdVE with
pioneering
KdVE with
the
only
our
rule:
of classes
set
translations
same
same
in his
the
waves
x
the
in
then
p,
Benjamin
where
kink
a
was
in
diffusion
nonlinear
a
is called
H1, satisfying
from
v
of standard
sense
At the
distance T.B.
waves
[48]:
Piskunov
for
to
for
C
JJu(-)-v(-+,r)JJHi.
inf
space.
solitary
O(wl,
in the
,ERN
of the
up to
of
be close
cannot
and
solitary
of
N.S.
kink
a
wave
following
the
then
waves
solutions
any
be
a
KdVE is invariant
smooth
and
u
R, equivalent,
E
7-
distance
the
stability
of
solitary
a
has
one
distance
u(t), belonging R(T (t), u(t))
with
any
R(T (0), u(O)) first
one-dimensional
for
historically
the
called
t > 0 is
b
exists
t > 0 and
Probably
t > 0.
any fixed
0 there
>
e
any fixed
X for
all
any
to X for
belonging
consideration, X for
to
complex
space,
usual
the
-r
E
R'
and
one-dimensional
7
E R.
To
NLSE with
cubic
fact,
this
clarify
f (s)
=
we
has
s
a
of solutions
V-2-w real at
t
exp
I i [bx
-
parameters. =
0 in the
(b
2 _
W)t]
Therefore, sense
of the
cosh[v/w-(x two
-
arbitrary
distance
p,
2bt)] solutions
cannot
from
be close
for
this all
4
t > 0 in
the
any two
standing
family
NLSE, of the
sense
of the
distance
40(x, t)
above
satisfy
in the
the
two values
to
correspond
of the
waves
corresponding close
they
if
sense
same
different
to
close
at
t
parameter all
p for
At the
to same
stability
of
By analogy,
W.
-
distance
of the
sense
nonequal
w,
t > 0.
definition
the
of V
values
0 in the
=
each
other,
time,
the
in the
p and
of the
sense
be
cannot
functions
of
distance
d. In the two
necessary")
O(x)
>
for
nonlinearity
a
0, that
Next,
for
In
Chapter
theory
For the energy
we
and, for
the
higher
problem,
tific
contacts
appearance
and of the
Roughly
literature.
(with respect d Q(0) > 0 dw
condition
opinion
that
stability
3 is
devoted
oo.
We present
that
"almost 0 and
=
speaking,
to the
distance
is satisfied. to the
respect
kinks
of kinks
distance
always
are
under
stable.
assumptions
objects
many
on
dynamical the
If
recurrence
construct
KdVE in the we
wishes
present
theorem
explains
measure
associated
when it is solvable
infinite
corresponding for
by
our
phenomenon with
the
the
equa-
for
dynamical partially.
conservation
method
of the
of invariant
measures
colleagues
and friends
for
that
contributed
sequence
to
observed
was
measure
in
according
stability
the
this
con-
is well-known
phenomenon
invariant
theory
application
which
by
the
of
inverse
associated
laws.
to
thank
discussions
present
case an
generated
measures
in
such
the
means
bounded
invariant
an
it
system
a
one
phenomenon
Fermi-Pasta-Ularn
have
we
on
direction.
invariant
applications
important
interesting
in this
open
of the
waves
and
new
a
constructing
attention
speaking,
Roughly
a
simulations, Poincar6
our
of
remain
of
Fermi-Pasta-Ulam
waves.
of
questions
solitary
of
stability
a
problem
the
have
stability
the
to
many
with
It is the
conservation
The author
lim 1XI-00
for the KdVE with
the
prove
We concentrate
equations. the
NLSE,
scattering
deal
trajectories
then
-+
however
we
of nonlinear
many "soliton"
system,
JxJ
These
By computer
tion.
with
4,
if the
widespread
we
satisfying
is stable
(and O(x)
sufficient
a
type.
as
physics.
of all
view,
present
physical
wave
of kinks
a
Chapter
systems.
with
Poisson
of
equations.
dynamical
the
is
be said
should
nected
general
part
non-vanishing
It
our
there
we
waves
in the
NLSE)
the
NLSE,
solitary
of
solitary
a
stability
of
point of
f
The last
type.
type
the
consider
this
function
NLSE
stability Q-criterion
general
of
we
Confirming the
KdVE and the
the
the
is called
Among physicists
p.
of
for
KdVE and to d for
the
p for
of the
cases
condition
with book.
all
his
them
have
the
useful
importantly
sciento
the
Notation
the
of the
case
otherwise,
stated
Unless
KdVE and
I for
(X1)
X
N
8'Xi
i=1
R+
[0,
=
For
E
=
positive
denote
positive
a
always
are
constants.
for
integer
NLSE.
the
Laplacian.
Q C RN
domain
defined
on
Lp(RN)
and
D with
Mp
=
the
L,(Q) (p : 1) is the usual i JUIL P(o) ff lu(x)lPdx}p.
norm
Lebesgue
lUlLp(RN).
f g(x)h(x)dx
for
(ao,aj,...,a,,...)
=
g, h E
any
L2(9)-
11al 1212
R,
E
an
:
00
1: a2
0 the
any
T); Hpn,, ,,(A))
that there
This that
global
of
solution
Hn-solution for
the
in
all
t
R).
E
periodic
case
following.
problem sense
continuation
a a
problem
of the
book is the
> 2
about
about
and
well-posedness
in this
Theorem any
is correct
it
wider
a
Hpn,,,
the
standard
global
unique
a
-3
i
u(-, t)
)
(A)).
KdVE.
periodic
depends
continuously
map uo
T, T);
with
is
on
there
for
H'-solution the
initial
from
continuous
In addition
Then
exists
a
data
Hpn,,(A) sequence
of quantities A
Eo (u)
A
I u'(x)dx,
Ej(u)
0
I
2
U2(X) X
6
3(X)
dx,
U
0
A
E.,,
(u)
12
[U(n)12 X
+ CnU
[U(n-1)]2 X
qn(U7
...
(n-2))
dx,
)U X
n
=:
2,3,4,...,
0
where
periodic
Cn
are
constants
Hn-solution
and qn
u(.,t)
of
are
polynomials,
the
problem
such
(1.1.1),(1.1.2)
that
for
(with
any
integer
f(u)
=
n
u)
> 2
the
and
quanti-
a
CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE
12
E,,(u(.,
Eo(u(.,
ties
regularization
following
Wt
1.1.3
f(W)W.'
consists
Proposition
(1.
3).
1.
At the
fact,
1.1.6
global
unique
(4)
+ WXXX+
step,
6W
first,
At
steps.
xER,
0,
=
X
E,,
consider
we
t>O,
are
the
(1.1.4)
e>O,
(1.1.5)
(x)
Wo
=
take
f(-)
to
the
limit
E
uo
function
satisfying
the
(0, 1] the problem (1. 1-4), (1. 1. 5) has n 1, 2,3,.... ([0, n); S) for an arbitrary
c
E
=
is,
of course,
of
exists
independent
an
differentiable
S there
(1. 1.4),(1.1.5).
problem
+0 in the
--4
infinitely
an
any
differentiable E
Coo
which
be
for
Then,
S and
belongs
statement
Let
infinitely
an
uo E
any
we
1.1.7
(1.1.3).
be
which
following
Proposition estimate
for
Then,
second
f(-)
Let
solution
the
get
we
Eo,...'
following:
the
estimate
junctionals
the
(1.1.1),(1.1.2):
problem
(x, 0)
a
e.
of several
W
and prove
i.
t,
on
Hn-solutions.
of the
+
depend
do not
of Theorem
proof
Our
t))
for periodic
laws
conservation
function a
satisfying
the
u(.,t)
solution
unique
In
interest.
00
U C-((-n,
n); S) of
the
(1.
problem
1.
1), (1. 1.2).
n=1
At the
third
Now
using
step,
II ( 00
P1,0(u)
=
)
dx1
2
dx
00
the
generates Proof
of
The system
Lemma 1.1.8
topology
follows
in
from 00
2
PM 1(u)
x
,
21
the the
we
Proposition
proving
to
turn
we
1.1.7,
Proposition
I
I
1
2
and
following:
the
po,,(u)
1
00
x21u2(x)dx -00
I
00
2
u(x)
dx
dxm
dm
dx-
[X2,dmu(x) ] dxm
-00
k=O
0, 1, 2,...
S.
space
(dM ) E
=
relations
U(X)
min m;211
Cl""
with
seminorms
00
:5
We begin
1.1.6.
1.1.3.
Theorem
prove
j -.
I
x
2(21-k)
u
2
(x)dx
+
d2m-kU) (dX2m-k
2
f
dx.
0
dx
n
(1.1-6)
2,3,4,...,
=
U0(X)j
=
Fourier
(1.1.7)
transform,
one
easily
can
show that
00
U
Wn E
([0, m); S),
C-
n
2,3,4,....
=
M=1
Taking
into
(1.1.3)
account
applying
and
Sobolev
embedding
inequalities,
we
get from (1.1.6):
[Wn ( ) 21
00
I
1 d 2 dt
2
-00
00
192Wn
+
-WX2
(194 ) Wn
dX
I (Wn
OX4
+
a4
00
Wn) f(Wn-1)
19Wn-1
dx
+
1)(IW(4) 12
Gronwell's 0 such
Let
us
now
JWn 12)
+
nx
obtain
21luol 122
< -
the
I
-,E
YX2
C2(f)(1
+
) (,94 )2 2
Wn
+
a IWn axl
9X4
'9X
2
(1.1.10)
estimates
let
112(p+l)
dx+
immediately
2,3,4,...,
t E
11 W j 122).
+
2
implies
the
n
,
[01to]-
us
(1-1.8) existence
(1.1.9)
2
< -
c(E,
I
=
3, 4, 5, induction
in
1, equation
00
2
nx
Now,
I JWn-1
(1.1.8)
....
W(1+2)
n
dx-
2
E
00
m,
a2Wn
[0, to] and n 2, 3, 4, By using the and we get: (1. 1.9) embedding theorems,
t
estimate
=
0XI
estimates
ax,
and the
0 is
trivial.
large the
also
d=-oo
I
X2mW2dx + C, (c, m), n
r
I
=
I and
=
For
the
for
W2xx]
K
large.
arbitrarily
example,
6
K)W2 + n
n
dx +
The terms
Pi of the
terms
C22
Pi of other second
kinds
kind
we
can
be
have
00
Pj :5 C + C
f
X2m(W2_,
+
large
the term
n
W2 n)dx.
-00
00
So,
we can
choose
the
K> 0
constant
so
that
e
f
2mW2x.,dx
X
n
becomes
-00 00
larger
than
the
sum
of all
terms
of the
kind
f
2z K
2mW2x.,dx.
X
Therefore,
n
we
get
-00
(X,
I d
2 dt
I 00
00
X2mW2dx n
< -
C(c, m)
1 + -
I -00
2m(W2-,
X
n
+
2)dx
W n
(1.1.12)
(GENERALIZED)
THE
1.1.
KORTEWEG-DE VRIES EQUATION(KDVE)
(1.1.11) follows from (1.1.12). Inequalities (1.1.9)-(1.1.11) immediately yield in the space C([O, ti(r-)]; fWn}n=1,2,3,... S). Also, the
15
The estimate
00
Tt
2
of the
compactness
sequence
00
f
I d
the
estimate
gndx
f
C3 (E)
dx
0.
some
5)
t
(1.
uniform
of the
1.
with
of
Therefore,
solution
a
for
for
an
C([O, T']; S) (where T' the condition Jjw(-,0;c)jjj
C
E
Then
C
any
these
with
respect
L 1. 6 be valid
(1.1.5).
Also,
that
3)
[0, T].
E
of the
to
E
c
(0, 1],
for
C([O, T]; S).
class
(1-1.4),
> 0.
t
uniqueness
of Proposition
0 such
>
and the
estimates, 1.
E R and
x
proved.
is
problem
condition
satisfying
t E
the
argument
exists
of two
[W(X, t; 6)] 2dx,
on
lemma,
assumptions
solution
of
T > 0 there
function and
a
function
creasing and
be
some
(1. 1.4), (1.
problem
depend
not
(1.1.4),(1.1.5)
Lemma 1. 1.9 Let the
C([O, T]; S)
0 does
Gronwell's
make
to
with
existence
00
problem
want
we
the
suppose
us
00
[W(X, t; 6)]2
the
to
of the
S). Thus, taking the limit problem (1-1-4),(1.1.5)
Q0, T]; S) (1.1.4):
equation
-00
1 W
se-
L2) S),
of the
let
class
00
d
where
the
Q0, t'(c)]; C([O, ti(c)];
in
compactness
Q0, t'(c)];
space
local
Therefore, space
S).
uniqueness
w'(x, t;,E)
solutions
oo,
-+
to
w(x, t; c)
to
converges
the
and
fWn}n=1,2,3.... converges E (0, ti] is sufficiently
(1.1.6),(1.1.7)
in the
(1.1.6)
by equation
>
w(x, t; 6)
and let
lw(., t; 6)12 p E (0, 4), is
nonin-
a
0,
R,
> 0
arbitrary
infinitely
differentiable
constants
C and p,
any
(0, T]
oo
0.
in
view
Then,
of
the
there
of H1
embedding exists
R4
>
into
0 such
that
C). for
Take any
an
6
arbitrary E
(0, 1],
sufficiently an
arbitrary
(GENERALIZED)
THE
1.1.
differentiable
infinitely and
such
p and
arbitrary
an
(0,T]), I I W(')
t
1
6)112
c
(0, 1]
E
the
conditions
:5 R4 for
all
and let
infinitely
an
00
f (a2W )
d
Tt
-
2
p,
we
dx
9X2
(U(4))2
dx
x
I
-
CIO
00
1
dx
x
00
(T'
term
to the
we come
be estimated
can
'X,
Tt
2
after
an
00
I
1 d
by analogy
estimate
2mW2dX
f
C3 + C4
whose
[0,T*)
point T*-O
t
t
0, i.
>
us
way
for
as
belonging above
to
Now,
the
we
turn
I,
any
m
-+
oo.
of the
infinitely
1.1.7
For each
n
=
and
Let
us
sequence
1, 2, 3,
...
9, is bounded
the
and let
R2
=
with ....
sup
us
,
where
R2 (C7
Pi
=
Un
f (.)
estimate
2 E H and 2
clearly
W3 E (0, oo).
We set
> 0
be
same
solution with
the
in H and we
>
R2 0
-
>
Let
is
with
x
(- 1, 1); R)
>
0 and
let
in H'
strongly
denote It
=
function
continu-
If,,(')jn=1,2,3.... (1.1.3)
T
the
as
solution
clear
that
0 is
given
by
also
R3
Sup
n
Then,
a
twice
C2((_M, M)
uo
T)
the
of
and let the
in
weakly
1,
T
uniqueness
by analogy
((- T, T); S) Un. fn and uo 0
the
some
any
in
arbitrary
an
(1-1.3)
E C-
f
proved
uniqueness
satisfying
to uo
with
we
can
proved
take
estimate
to
(x),
ul
too.E1
arbitrary
take
converging by Un (X, t)
JjUnjjj,T)jn=1,2,3
Let
converging also
be
can
6)
The
be
can
and
proved,
functions
taken
JR2(CIP)
t < 0
considering =
for
(1. 1.4),(1.1.5).
T > 0
limit
a
problem
existence
1.1.3.
f () satisfying
problem
sequence
Theorem
p
of the
any
is
T* +
of
proved.0
Lemmas 1.1.9-1.1.12
+0 in the problem
differentiable
1,2,3,.... a
C([O, T); S)
E
c --+
is
exists
Thus,
[T*,
of time 1. 1.6
Due to
Proposition
C and
C S be
1.1.7.
domain
function
=
an
half-interval
w(x, T*; E)
data
interval
of
(1. 1.4),(1.1.5),
the
there S.
space
initial
So, Proposition
t in the
proving
to
on
the
C([O, T); S) for The (1.1.4),(1.1.5).
any fixed
constants
same
as
Thus,
of
sequence
Un 0 1 n=1,2,3....
1. 1.
class
differentiable
a
for
of the
with
problem
u(x, t)
limit
construction.
(1. 1.4)
existence
half-neighborhood
results,
of the
sense
0, solv-
t >
the
onto
right
above-indicated in the
Proposition
solution the
S for
the
to
contradiction.
a
problem
the
due
all
Suppose
arbitrary
an
for
t; c) of the problem
be continued
can
on
global,
the E S.
uo
w(x,
proved,
be continued
equation
prove
a
been
of this
let
S-solution
E S understood
for
by taking solution
ously
get
of
of this
n
Then, ul
=
now
existence
obtained
f
T*.
solvability we
e.
Let
the
be
cannot
=
get the local 6
and
Cauchy problem
the
already
imply
Indeed,
corresponding
has
El
immediately
(1.1.4),(1.1.5).
the
w(x, t; E)
lim
1.1.9-1.1.12
problem that
uniqueness
of time the
and
proved.
is
the
Lemma
I I Un0 1 12-
n
R4
=
R4(R3)
where
the
function
R4
R4(R3)
> 0
is
(GENERALIZED)
THE
1.1.
by
given
Then,
Lemma I.1.10.
jju,,(-,t)jjj For x
t
[-T,O)
E
these
and t
-x
--
(I.1.1).
equation
t)112
5
W4)
(-T,T).
t E
Therefore,
have
we
21
1.1.10,
and
(1.1.25)
change
by the simple
be obtained
can
for
of variables
t > 0
00
0"
I
JjUn(')
and
estimates
in
-t
--+
due to Lemmas 1.1.9
R2
!5
EQUATION(KDVE)
KORTEWEG-DE VRIES
d
(Un
2 dt
Um) 2dx
-
=
(Un
-
Um)(fn(Un)Unx
-
fm(um)um.,)dx
-
-00
00
I
J(Un
Um)[f(Un)(Unx
-
Umx)
-
+
Umx(f(Un)
f(Um))+
-
0.0
+(fn(Un)
f(Un))Unx
+
Umx(f(Um)
fm(um)jjdx
-
00
f
C(T)
(Un
um)2dx
-
+ an,m)
00
where
an,m
--->
+0
n,
as
convergence
of the
Due to the
estimates
m
sequence
let
us
take
fUn(*) t)}n=1,2,3....
+oo and
by analogy
fUn}n=1,2,3....
in the
for
t < 0.
These
TI; L2)
C([-T,
space
yield
estimates to
the
u(x, t).
some
(1.1.25),
u(-, t) Indeed,
---
weakly
t E
[-T, T].
2
E H
loss
of the
(1. 1.25),
Due to
generality
I JU('7 t) 112:5
and
tE[-T,T].
H2 , hence
in
compact
the (without Therefore, JUn(') t)}n=1,2,3_.).
subsequence
u(-, t)
JjU(',t)jj2<W4,
and
arbitrary
an
is
2
E H
it
the
contains
sequence a
weakly
that
accept
we
liM illf
(1.1.26)
I JUn('7 t) 112
it
0
be
Un(',t)
in H for
JUnz,&i t) 122
any
2
t E R.
u(.,t)
--+
in H2
strongly
Due to Lemma 1. 1. 13 and the 2
by analogy.
proved
Further,
n
above
we
1 Unxx ('1 0) 122
as
-00
--+
n
--+
have
from
C((-T,T);H').
I JU(*) t)
(-, t) (1. 1.16),(1.1.18) un
fIf( Un( 3))Un n
in
oo
then
oo,
arguments
6 0
as
I
3
-
Un(* t) 112 i
---+
u(-, t) with
-9)Unxx(
i
as c
-9)
=
+
--+
n
0
0
CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE
22
0"
(.,s)ldxds+'
5
+
6
nx
f f f- (U- (" 0) Un'x (',
-
6
t)
-00
-fn(Un (.' 0))U2 nx(*, 0)JdXWe want n
show that
to
Consider
the
right-hand
the
Obviously,
oo.
--->
the
for
last
R(f,,,u,,) (1.1.27)
side in
term
3
0
f"(u)uxuxxj
dxds
the
R(f,u) is
as
valid.
3
-
-00
00
I [fn (Un) 0
3
f"M]
-
3) + Ux
3
ff(U) [Unxx (Unx
UnxUnxx +
_
U3X(Unxx
uxx)]Idxds.
oo
-
Due to
embedding
C((-T,
T)- JJ2)'
and
its
weak
this
equality
(1.1.27)
theorems, its
to
--+
Let
Un
I
by
R(f, u) 0)
U
t) 12
u(., t) as
n
-4
in H2
1 2
(1.
problem
1.
1), (1.
1.
all
2) by +
1vt
an
arbitrary
Then,
we
above
problem,
C ((-
T, T); H')
get
Wn(.) 0
addition
2
the
that
taken
and, ___
the
any
f
< -
1liminf
2
n-c*
1 2
.
-+
the
right-hand
side
of
the
right-hand
side
of
that
of the
integral
of
and
n
Then,
oo.
lim inf
S.
we
in
view over
Therefore,
of
em-
(0, t)
in
indeed
0)12
I Unxx
have
+
n-oo
+
R(f, u)
R(f, u).
considerations
+ WXXX=
(1.1.28) also
are
fixed
W(-, t)
0, C S
infinitely
of =
U(., t) strongly
1 U x (.' 0) 122
in in
observe
integrand
n
-
I Wn o}n=1,2,3....
sequence
for
we
in
Lemma 1. 1. 13
valid
if
we
change
the
following:
f (7v)u)x
with
as
1 Uxx (.' 0) 122
above
the
sequence
term
independent
I Unxx (', t) 12
lim inf
that
addition,
In
of the
constant
(*1 0) strongly
n-oo
observe
u(-, t) following expression
from
oo.
_
we
the
fUnjn=1,2,3....
sequence
The other
oo.
value
positive
n
exist
oo
in
E
>
H2).
Q-T,T]; 0 and
sequence
a
as
k
as
k
--+
and
oo
-*
oo
6-
Itn}n=1,2,3...
sequence
in H2
from space
Lemma 1. 1. 16 that
H-1 for
solution.
Let
defined and
U'
=
ul
in U1
I
'0
-
t
any
E
of the
solution
00
1 d
-4
in H2
one
So,
converging can we
easily
get
a
to
some
prove
as
in
contradiction,
proved.0
generalized
(1.1.1),(1.1.2) t E [0, T2)
n
there
U(',tn)112
-
-+
7
is
follows
easily
It
and
invalid
of the
subsequence
a
as
that
JjUn(* tn) Let
u(.,t)
(., t) an
-
7
problem
and
interval
f (u (-, t)) u', (., t) uxxx (., t) (. t) [-T, T] and, thus, we have proved the ut
U2
('7 t)
Let
be two
of time
(-Tl,
generalized
T2)
where
U2: 00
2 W
(X, t)dx
W(f(U1)U1x -00
-
f(U2)U2x)dx
us
prove
solutions
T1, T2
the
of the > 0.
We
(GENERALIZED)
THE
1.1.
jA j (
)2
d Xn
2
+ CnU
KORTEWEG-DEVRIES
(
)2
q,,
u(x),...'
polynomials,
such
d Xn-1
-
EQUATION(KDVE)
dx
dx,
n-2
25
n>2,
0
where
Cn
real
are
differentiable
period oft, of
i.
e.
the
the
quantities
the
junctionals
statement
with
f(u)
readers
consisting
weakly
to uo
infinitely period
and t E
Then,
all
strong
I
integer
Hpnr (A)
in
I
as
2,
result
(1.1.1),(1.1.2)
questions,
(see
is obtained
of the
Hn,,(A)
uo E
and
periodic
(x, t),
Let ul
oo.
--+
proof
the
in
as
1, 2, 3,
=
in
I
a
with
x
1, 2, 3,
=
we
refer
Additional
1.1.3,
one
can
fU0(1)}1=1,2,3. .
sequence
period
the
A converg-
corresponding
be the
...,
periodic
in
show that
for
problem
of Theorem
and
...
JU(*) 0)12
=
that
and
ul 0
now
--+
is
and,
A
I
A
3(x)
dx
U
f
and
0
weakly
on
En (u (.,
addition,
here in
Hpnr(A)
Suppose
the as
this
0))
I
---+
any
R,
strongly
and
continuous
CnU (dn-lU)
space,
=
we
of the
with
x
the
any T > 0
to
2 _
gn
I-00
Hpner(A);
in addition
:
0 be
(U'. .'
arbitrary.
and the
dn-2
u
dXn-2
Since
the
functionals
dx
have
En (ul (., 0))
lim inf
f U1 (X, t) 1 1=1,2,3....
5 C1-
Hpnr(A)
on
:5 C1
sequence
weak in
JjU&)t)jjHPn r(A)
I-00
equality
strong
(x, t)
dXn-1
0
continuous
u
t E
:5 liminf
& are obviously
Eo,...,
limit
a
for
HPn, ,r(A)
Uo in
jjuj(-,t)j1Hpn,,(A)
max
tE[-T,71
there
T); Hpn,-r'(A))
C((-T,
in
functionals
strongly
this
other
to
(-T, T)
Let
in
the
H' -solutions
problem
of the
devoted
is
functions
1JU(*)t)11HPn_(A)
are
book
>
n
solutions
U(*) t)12 for
with
independent
and
for periodic
laws
integrability
where
differentiable
differentiable A.
complete present
determined
are
infinitely x
u.
=
literature
arbitraiy
infinitely
of
the
t)),...
in
chapter).
take
us
since
arbitrary
an
periodic
conservation
are
f (u)
with
for
that
problem
E,(u(.,
to the
corresponding
this
to
Let
and,
u
the
to
1), (1, 1.2)
1,
the
&,...
Eo,...,
is related
=
remarks
ing
(1.
problem
This
are
of u(x,t) Eo(u(.,
solution
A,
and qn
constants
takes
place
if
En (u
(., to));
and
only
if
u&, to)
--+
u(., to)
oo.
that
En (u (.,
0))
>
En (u (, to)). 7
(1-1-31)
CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE
26
IUI,}I=1.2,3....
Let
argument
strongly
Hp' JA).
Hpn,,(A)
in
(-, t), (1.1.31),
of the
I
ul
of
...
get
as
differentiable
converging
functions to
of the 1
as
-4
(1.
1.
infinitely
differentiable
1) satisfying
ul
(., to)
lim inf &(u,
(-, 0))
periodic
ul, (-).
=
solutions
Therefore,
due to
have:
we
we
e.
infinitely A and
of
sequence
equation
E,, (u (-, 0)) i.
period
00 u(-, to) autonomy of equation (1.1.1), the function CQ-T, T]; Hpn ;'(A)) and, for any fixed t E [-T, T], weak
corresponding
1, 2, 3,
=
the
in view of the
in
strong
of
sequence
with
Then,
limit,
is the
arbitrary
an
periodic
E R
x
in
u(-, t)
be
E,, (u (-, to))
>
contradiction.
a
Thus,
=
1-00
for
ul(.,
any t E R
t)
(u (., 0)),
! E,,
u(., t) strongly
--+
HPn,
in
oo.
One
that,
if
again
then
ul(.,
t)
t)
ul(.,
complete analogy that u(-, t) E C([-T, T]; Hpn,,(A)) and u. C Hpnr(A) in Hpn,,,(A) I fUoj1=1,2,3.... as uo strongly in T > 0 u(., t) strongly C([-T, T]; Hpn,(A)) for an arbitrary the
-+
corresponding
are
proof
periodic
En(u(-,
t))
Hpn r(A).
I 1 .5 about
follows
The
In this
the
-
from
Theorem
1.2
the
1.1.5
time-independence
continuity
completely
is
nonlinear
section,
we
prescribed
consider
We shall
results
iUt + AU +
f(IU12)U
initial
complex
plane
f(JU12)U
tion
for
the
is
k times
the
complex
n)
x
En
t)),
the
on
...,
space
=:
0,
=
Uo
(NLSE)
equation of solutions
existence
N,
of the
NLSE
(1.2.1)
t E R
E R
x
(X, 0)
(X).
(1.2.2)
operator
A in
equation
operator
-D.
Here
function
continuously
f(JU12)U
:
linear
differentiable
00
U C'((-n,
Eo,...'
The state-
Eo (u (.,
quantities
functionals
the
on
two-dimensional
the
as
Laplace with
it
the
C
H'-solution uo.
data
identifying
smoothness
of
the
of the
Schr6dinger several
understand
sense
of
where
proved.0
U
ized
a unique u(-,t) global periodic depending on the initial data
1.1.3,
oo,
--+
problem
of the is
continuously
of Theorem
ment
Hn-solutions
of Theorem
problem
of the
and
-->
As in the
with
by
prove
can
R2)))
(-m, m);
if
f(JU12 )u
as
a
(1.2.1)
we
C
1
space
R2,
(we
write
map from
shall
conditions
accept
Considering
C.
)
we
say
this
in
2
R
general-
the
in
into
that
the func-
the
f(I
case 2
R
is
U
12 )U
E
k times
m,n=l
continuously
differentiable.
To formulate
jxj
--+
oo,
we
need
a
the
result
on
following
the two
existence
of solutions
assumptions.
of the
NLSE
vanishing
as
THE NONLINEARSCHR6DINGER EQUATION(NLSE)
1.2.
(fl) f (s),
and
Co N
where
such
2,
>
s
be
0,
-
differentiable
continuously
a
a
real-valued
1),
where
be
(0, p*
and p E
> 0 =
f(JU12 )u
Let
function p*
if
N-2
of
in the
3 and
N
the
p*
argument
(9
Co(I
i)u
>
Let there
Remark u
C
exist
0 and pi
>
a
Under
1.2.1
iU2
U1 +
-=
exist
arbitrary
1 is
(0,
E
+ '
N
)
JUIP), such
U
for
(1.2-3)
E C.
f(S2)
that
0 t
U1
('7 t)
fIUI(*7
t) 12 :5 C1
U2
-
S)
U2(*7 8)12ds7
-
0
hence
ul
and the
(-, t)
Let
is
right-hand
side
of
existence
of
infinitely
T(R) (1.2.4),
=
of
For t < 0 this
t > 0.
solution
a
the
an
of T
existence
for
of
prove
us
f(JU12)U
that
(*, t)
_= U2
uniqueness
(Gu)(t)
=
solution.
a
differentiable
>
e
-itDUO
IluoIll
if
I
+
I
Let
any
R, then
Suppose
integer.
be
I
>
For
the
us
now
show the
G in
operator
the
t
e-i(t-s)D[f
(jU."9)j2)U(.'
s)]ds,
0
the
maps
set
MT
mutes
with
C([-T,TI;Hl)
E
u(O)
:
G(MT) C C([-T, by applying embedding
Indeed,
itself.
into
Ju()
=
uo E
jju(-)jjj
H,
T]; H') and,
clearly
e",
=
theorems
we
21luolll}
:! , the
since
-2-
operator
get from (1.2.4)
ax
for
com-
u(-, t)
E
MT: t
jj((;U)(t)jj1
JIU0111
:
+I7j(j u(-,s)jj )ds, 0
where
-yi(s)
implies
is
continuous
a
the existence
function
positive
of theaboveT
>
of the
argument
s
Iluolli,
0, dependingonlyon
This
> 0.
inequality
G(MT)
for which
MT. Now let
the
map G is
theorems
for
us
show the
a
contraction
of the
infinite
differentiability
and the
existence,
set
any R >
MT, of
0, of
Iluolli f(JU12)U,
if
a
constant
:5 R. we
Ti
Indeed, easily
E
(0, T]
using
derive
t
II(Gul)(t)
-
(GU2)( )III
:5
C21 jjU1(,9)-U2(8)jjjds, 0
U1(*,t),U2(',t)EMTj,
such
that
embedding
c
CHAPTERL EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE
30
where
the
C2
constant
depends
> 0
only
Iluolli,
on
and therefore
such
T,
constant
a
exists.
Since we
have
that
for
the
interval
of
fixed
a
proved, any
E
H' there
uo
the
on
(-T, T)
(1.2.4).
Since
the
not
length
by
and
of the
I Ju(-, t) 111
limsup
T >
there
t
is
of
of the
TF
existence
of
again uo H'-solution
unique
Tk+,
be the
and
0
7
i
exists
I JU(* t2) 111)
on
for fact
....
there
tE[-tl,t
Hl-solution
complete
the
verification
direct
C([-t1,t21;H'). fjjUn(-,ti)jj1}n=1,2,3
i
get that
obviously
Mare
the
since
shall
proved
be
can
we
by analogy
0 and
The
in
sequences
[-t'j, 0] (resp., on [0, t2]) and, as above, (resp., in Q0, t']2; H1)). Continuing this finite
constant
=
oo
if tj
Hence, numbers
C5
--+
of the
depending
large
sufficiently
and
convergence
respectively. E
oo
u(.,t)
--+
particular,
all
as
u,,(-,t)
therefore in
+0
--
functionals
on
of t of the
independence
the
space
quantities
H'-solution u(-, t) of the problem P(u(., t)), E(u(-, t)) and M(u(., t)) for an arbitrary > also the these above arguments to are quantities 3), according (1.2.1),(1.2.2) (1 of t if u(-, t) is a H'-solution of this problem. independent the under we get Finally, assumption (f2) using inequality (1.1.15), 00
1 F(ju(x)
12 )dx
0
such
that
as
Fix
for
all
t
--+
arbitrary
an
t
Xk because
0 in
:
Itl
6
< I
> 0.
the
+
JI31)
one
is valid
tends
to
zero
integral
improper zero,
for
consequently,
and,
oo
--
which
(1.2.8)-(1.2.10)
t > 0 and
any
,t[Gtuo]
i(4ri)
=
to
right-hand
any bounded
58-t [Gtu0j
uo'(x VZ_
2vft-z
-
side
interval
not
and
is determined
at
+
side
any bounded
t from
in the
term
in t from
2vlt-z)
u"0 (x +
2
respect
derivative
00
right-hand
in the
term
second
the
R the
E
x
with
uniformly
converges
first
the
uniformly
oo
c --+
as
due to estimates
Since
containing
jxj
as
)dz
0 00
i(47rit)-
e
JEWL, U/1 (y)dy
=
0
a2
-[Gtuo] X2
i
Co
for
t < 0
at
t=o
=
a
[Gtuo]
=
there
lim
exists
These
0.
L
first
statement
(1.2.14),(1.2.15) Now
equation
is we
homogeneous
any
X3_SolUtion
that
In view
any
of the
X'-solution
only
X'-solution
the
=
of
iu".0
Hence,
imply
also
arguments
of
arguments there
exists
that
=-g(u(xt)). (1.2.7)
equation problem
the
above
arguments,
the
satisfies
problem
Lu=O,
xER,
it
suffices
(1.2.14),(1.2.15) to
that
prove
satisfies the
linear
problem
U(X, 0) has
at
[Gtuo]
above
proved.
prove
(1.2.7).
that
a
I
I ijGt_,(g(u(-,s)))ds 0
our
due to the
Further,
u".0 t-0
t=o
L[Gtuol
iu" 0 and
2
a2
t
Thus,
relation
of this
proof
the
earlier,
noted
we
by analogy.
iGtu" 0 and therefore
[Gtuo]
As
t > 0.
obviously
0, then
-12jGtuo] at
0 for
=
be made
can
If t
-2-
L[Gtuo]
indeed
that
so
trivial
solution
of this
problem
u
=-=
in the
0 from interval
=-:
C (I;
tEI, 0
X3)
of time
Let
.
Simple
I.
0')
d
Tt
j
I ux (x, t) I'dx
=
us
0
suppose
that
calculations
u(x, t)
is
show that
a
Concerning in
the
For
simplest
results
the
on
[102]. f (s)
case).
with
sP
=
NLSE
essential
an
periodic periodic =
and
u)
we
no
uniqueness
KdVE and for
of
As for
the
phenomenon
is unknown
for
the
this
[69,70]). blowing
For up
[38], f (s)
this
in
as
solutions,
NLSE) to
paper
NLSE with
=
the
sP with too.
paper
result
is p
recall
proved
that
we
1.2.10
well-
(see,
one
of the
considered
only
first
are
proved
[91] stating
Cauchy problem
of the
non-smooth
in
for
for
by
J.
problem
Bourgain in
the
solutions,
proved
for
have
(see, in
of the
one we
rigorously proved
[16] (see
[17])
the
for
to
the
where
the
(with Is IP.
usual
the
f (8)
like
it is known for
considered
vanishing
devoted
literature
is
problem
data
also
are
x
there
of the
initial
nonlinearities
Although is
with
whole
superlinear
[38].
data,
initial
well-posedness
the
review
have
it
of the
smooth
more
of Y. Tsutsumi
of the
and the
justified
2-, N
or
have
we
Theorem
with
of the blow up of we
H'
L2.
periodic
L2-solutions
KdVE. Above
and followed
subject
technique
from
possibility
the
uo
result
investigations
the
of
from
1.2.4
form is contained
investigations
lot
well-posedness
the
consideration
the
mention
and
general
in its
(we
important
data
between
Having
problem,
existence
f (u)
(with
x oo.
--+
-) (0, -N
N = I for
Theorem
1.2.7
the
initial under
difference
in
IxI
P E
equations
the
Proposition
with
a
data
of Hl-solutions
mention
some
(1.2.1),(1.2.2)
For
With
1.2.4
are
initial
with
existence
especially
We also
there
equation,
[33,37,69,70,79,88]).
one-dimensional
the
this
of Theorem
proof
the
Cauchy problem
of the
example,
for
NLSE,
the
[45].
paper
posedness
as
EVOLUTIONARY EQUATIONS. RESULTS ONEXISTENCE
CHAPTER1.
38
=
NLSE but
the
simplest
results
on
formal
presented
only
example,
[90] and, also,
paper
[90]
that
a
there
exist
Chapter
2
problems
Stationary As
noted
already
have
we
standing
for
sentation
AO 0
is
a
real-valued
supply
their
_
(IJ.0-1)
Usually
the
problem
N
1,
=
with
following
jxj
as
11. 1_00
QC
RN is that
Suppose function
and
problem
pairwise
By analogy,
a
oo
N
02)0
f(X, 02)
as
it
will
it
Q.
It
has
solutions.
then, as
domain
3 and in
is
=
(11.0.1)
O(x).
O(x)
equation,
of this
Rlv,
E
(11.0.2)
conditions
jxj
as
infinity
the
on
vanish
--+
oo,
i.
(11.0.3)
be reduced
assumptions
natural
KdVE under
can
for
e.
0.
(11.0.1),(11.0.3)
bounded
>
k(x) > 0 (11.0.4),(11.0.5)
different
starshaped,
now
x
some
=
the
for --+
If
is
f
proved
be shown
=
IOIP-',
in
[741,
further
P.E. Zhidkov: LNM 1756, pp. 39 - 78, 2001 © Springer-Verlag Berlin Heidelberg 2001
X
equation
to
(11.0.1)
(11.0.2),(11.0.3)
and
on
with we
E
Q,
0
with
also
(11.0.4)
O(X),
(11.0.5) a
smooth
sufficiently
k(x)jOjP-1,
=
=
0,
=
p >
that
if p E
positive
in
p >
7Nq j2,
known
solution
a
0,
=
Olau
the
the
to
problem:
WO+ f(X,
_
0,
=
solutions
problems
the
similar
AO
where
repre-
NLSE leads
generalization
a
with
that
waves
solutions
Along
too.
the
consider
solitary
finding
of
of these
behavior
RN, 0
E
consider
(11.0.2)
and
O(X) The
x
02)0
WO+ f(X,
suppose
we
0,
=
We also
function.
equations
solutions.
f(02)0
wO +
-
AO and
e"'O(x),
general
of the the
into
R,
E
w
equation
stationary
Here
u(x, t)
=
substitution
the
Introduction,
in the
waves
N-2
this
problem
(see
Example
1,
(1, N 2)
and
2
Q and w
has 11.0.1
boundary.
where
an
w
infinite
0 and no
k(x)
the
a
C'-
0, then of
sequence
domain
nontrivial
and
is
Section
fl
is
solutions. 2 of
this
chapter), critical
the
fying
condition
f(02)
lim
if
and
infinite
an
JxJ)
=
been
have
problem
(11.0.2),(11.0.3),
voted
it.
to
simple
k(.)
11.0.1
tends
rapidly
Let
to
zero
as
f (x, 0')
above,
under
similar
:N 22
p
0 and
1: I
N > 3,
x,
C, :5 k(xi)
+oo for
is
[99].
from
C'-function,
a
has
[1,77,78]
investigations
of
specific
some
radial
with
deal
that
mention
we
taken
Example
lot
a
are
f(.)
function
and
not
(11.0.1),(11.0.3)
We illustrate
example
is
there
problem
of the
lim
f (.) have been intensively
shall
we
f
problem
the
example,
interesting
many
book
present
literature,
In the
see,
nonlinearities
obtained
in the
However, lutions
(on
if
speaking,
(resp.
for
N+'
=
1951-oo
different
subject
p*
f(o 2) 0 (or f(x, 0 2) 0)
subcritical
subcritical
a
pairwise
of
this
superlinear
with
0 and
>
w
sequence
solutions
Problems there
with
N-2
1, and
p >
The exponent
called
is
OP',
=
nonlinearity
Roughly
+oo.
=
(0')
3, f
N+'.
the +oo
=
101-
(or (11.0.1),(11.0.3))
with
f(02)
Jim k61-00
N>
0,
usually the
we
O(x)
have
look
important
for
radial
results
-=
0.
solutions
by B. Gidas,
of the Ni
problem
Wei-Ming
(11.0.1),(11.0.3). and
L.
Nirenberg
In
41
[34,35]
be mentioned:
should
solution
problem
and of the is shown we
also
[9]
on
above
in the
assumptions
authors
under
the
with
assumptions
of
by
problem
a
function
of
general
similar
of
this
is not
and
P.L.
Now
so.
from
Lions
(H.0.1),(H.0.3)
to
x
As it
is radial.
type
sign
positive
independent
f
Berestycki
H.
arbitrary
an
a
alternating
with
result
that
proved
ball
a
solutions
solutions
general
very
a
have
in
proof)
a
of radial
existence of
for
papers,
(without
present the
these
(H.0.4),(H.0.5) (11.0.1),(11.0.3)
problem
of the
under
kind. 00
Theorem
11.0.2
9(-)
Let
U C((-n,
E
n); R)
be
odd,
3 and let
N :
n=1
(a)
'()
0 < lim inf
(b)
lim inf
(c)
there
'()
Jim sup 9W U
where
(0, +oo)
71 E
p*
again
-
exist
+00;
0
up
1-1-+00
=
f g(s)ds.
-2
0
Then,
problem
the
AO has
a
countable
In this
chapter,
qualitative
u(jxj),
=
of pairwise
different
we
shall
this
prove
establish
same
form
(N
2)
-
N,
solutions
radial
Theorem
011xj_.
E R
x
and
=
0
solution.
radial
positive
a
particular
in several
and
cases
study
the
of solutions.
also
we
in the
u
set
behavior
Now, taken
g(u),
=
as
Pohozaev
the
for
identity
(H.0.1),(H.0.3)
problem
the
[87]:
in
1 IV012
dx
=
-(N
1 Og(O)dx
2)
-
=
I
N
RN
RN
G(O)dx
RN
0
g(o)
where
Wo
=
f(02) 0
_
G(0)
and
-2
=
f g(s)ds.
equality
A similar
also
was
0
[74]
obtained
in
bounded
and
for
the
the
equation.
the
above
For
equality
equation
(11.0.1)
multiply
the
RN with
only
the
the
the
N 2 (N 2
independent
this
of
w
the
>
in
O(x) is
valid,
.5)
taking
into
integration
80('ax,),
sum over
i
1, 2,...,
case
the
function
1,
are
sufficient.
(11.0.6)
=
our
on
p >
of the
in
domain
one
result
may,
over
N and
first,
RN
integrate
and,
Q is of
the
f(o 2) To
in
get
multiply second, the result
by parts.
particular =-
jolp-1,
=
when the
case
Of course,
x.
account
further xi
the
assumptions
f(02)
0 and
in
of
additional
integration
identity This
3).
needs
by
solution
sign.
is
case,
with
equation use
The Pohozaev
change
f
example,
by 0
trivial
(11.0.4),(11.0
one
in
same
over
problem
function
(H.0.1),(H.0.3)
problem
p >
the
0 if
for
yields the
that
function
example,
for
(H.0-1),(H.0-3) NG(0) + (N 2)0g(O) does w > 0 and 101P-1 f(02) the
problem
-
=
has not
with
CHAPTER2.
42
Existence
2.1 first,
At
we
solitary
consider
u(x, t)
substitution
of solutions.
O(x
=
for
waves
ct),
-
-Wol Assuming
0(oo)
that
following
the
to
we come
a
NLSE with
a-
to
The
1.
=
equation
0.
=
0"(00)
and that
constants,
are
the
N
0,
=
equation: +
7(0)
0"
+
(II.1.1)
a
=
0
7(0)
and
-wa-
=
the
KdVE leads
the
into
a+ and
-wO with
KdVE and for
f M01 + 01"
+
where
a,
=
An ODE approach
the
R,
E
c
STATIONARYPROBLEMS
f f (s)ds.
is clear
It
that
substitution
the
general
of the
a-
for
representation
of
First
of this
solution
onto
(11.1.1) is bounded, 0"(-) is bounded, too,
equation
derivative
similar
a
However,
is
a
on
a
continuously
equation.
it
to
make
a
function
and
a
then
[a, b),
of the
follows
it
therefore
point
from first
the
where
this
if
equation
also
Setting
bounded.
Oo
0(a)
=
+
Oo, 0'(b) similar
Cauchy
the
00,
=
we
reasoning Let
fj(0)
problem
immediately several
=
for
get
times
f 0'(x)dx
the
0
second
solution
and
00
0'(a)
=
+
f 0"(x)dx
and
a
(II.1.1)
with
In this
statement.
this
specifying
not
can
b
equation
our
it
solution
of this
a
considering
then a
that
0'(.)
derivative
b,
2.
takes
and
real
parameter
_
following
the
g(y)
=
with
f (y2)y
0y
initial
obtain
First
we
of the
functions
0'(w,x)dx KdVE
2))]
2A 2t _
1)(v
and
+
2A2
w
as
family
one-parameter
+
vanishing
One
(v+l)(v+2)
easily
can
a
point 0
d.
In view
C,
By points
constant
=
we
obtain
i
I + 1
k
11.1.3,
(T;ml+l )2+ C2-
sequence.
a
of Lemma
statement
:5 C,
first
exists
1
the
1, 2,...,
C
1,
estimate
(11.1.9)
of extremum
lie
the
>
0 such
m=
that
1, 2,3,....
and
from the
since
the
outside
0
EXISTENCE OF SOLUTIONS. A VARIATIONALMETHOD
12.2.
the
Thus,
function
y(ro)
than
than
I roots.
the
on
At the
the
time,
same
yo and
parameter
definition
in view of the
imply
0
==
less
no
of solutions
dependence
continuous if
V has
the
theorem
the
on
that
y(ro)
V cannot
have
fact
the function
of -go that
49
=A
0
more
I roots. Let
that
prove
us
-9(r)
lim
Suppose
0.
=
this
is
the
not
Then,
case.
either
r-00
solution
the Zn
be
only
negative
for
can
number y E
first a-,
=
0) U(O, a,).
Thus,
contradiction
implies
of the m
y(r)
lim
=
y-
by
implies,
(11.1.4),(11.1.5)
0, and Theorem
H.1.3
theorem
the
than
more
on
parameter
I roots.
completely
is
domain
sufficiently
the
on
have
cannot
in the
for
negative
is the
since
case,
extrema
is
energy
problem
solutions
second
have
E(r)
energy
of the
negativeness
values
that
the
should
it
'g(r)
solution
the
of
which
asymptote
an
In the
r.
of extrema
sequence
a
has
P(r)
energy
1,
has
it
or
function
argument to
case
of solutions
large
sufficiently
this
in
But the
too.
the
of the
V is equal
solution
dependence for
that
values
r
of this
Hence,
a,.
=
large
of the
of r,
y
or
large
graph
the
case,
sufficiently
values
continuous yo,
y
of roots
(a-,,
large
the
In
sufficiently
for
V is monotone
+00-
-4
This
proved.0
r-oo
Existence
2.2 In this
section
the
to
we
shall
of radial
existence
f(02)
case
of solutions. consider
of the
jolp-1,
p > 1.
=
application
an
solutions
AO
A variational
wO
=
consider
we
101P-10,
-
Our result
the
on
Theorem
solution
We note
ul
that
right-hand
a
side
similar
sense
to
Theorem
tinuously
r(.)
ul(r),
'(rV)lr=r(,,)
=
point
a
N > 3 be
0,
where
r
=
takes
of the
equation
of
Loo
-
we
present
H be
Let
real-valued
Then, S,
jxj,
with
place
problem N
E R
(11.2.1)
,
(11.2.2)
a
solution
and,
precisely
1 roots
for
1
any on
g(o)
kind
(1, N+2). N-2
Then,
half-line
the
for
the
1,2,3,...,
=
(11.2.1),(11.2.2)
problem
the
general
more
and p E
integer,
for
then
if
a
below
real
are
Hilbert
functional differentiable
continuously
0.
vo E
attention
a r
> 0.
with
functions
g(O)
Theorem
11.2.1.
the in
a
101P`0.
which
a
the
proving
our
0.
radial
positive
result
11.2.2
be
>
similar
differentiable > 0
Jr
critical
Lo
has
=
Two results
Let
Let
to
We restrict
following.
is the
(11.2.1),(11.2.2)
problem radial
H.2.1
the x
=
existence
methods
(11.0.1),(11.0.3).
problem So,
of variational
method
the
Y(h)
space
in
j(v) -0.
when
proving
with
a
H, and S
function
functional
Ih=r(vo)vo
used
=
on
J(r(v)v)
norm =
f
S such
11
-
11,
h E H that
considered
be
J :
for
I IhI I any on
a
con-
1}.
=
v
E
S has
S a
Proof
is clear.
By conditions
Further, such
consider
-y(s)
that
E S for
d
0
arbitrary
an
(-1,
E
s
of all
consisting
Thus,
E L.
w
Remark
by
sults
S.I.
J(r(vo)vo),
H.2.3
The second
Theorem
of H1 with
Lq
is
E
the
compact
for
In
0.
=
is
a
Let
product
;
N-2
function
continuous
For any g E
1)
into
H
r(vo)-j'(0)
+
space
H
>=
>
subspace
the
=0
I
any
E H. 0
w
simplification,
sufficient
for
goals,
our
of
re-
fh
=
for
jxj
as
at
Clearly,
point
the
set
Thus,
0
x
C07,
=
Then,
h(jx 1)}
the
h E
we
H,'
0 and
there
of all
vanishing
exists
that
-->
except element
oo.
C000
from
into
unique
a
each
Ix I
as
functions
radial
subspace
of H,1
everywhere
accept
can
be the
embedding
RN and continuous
in
any
h
:
arbitrary
an
oo.
---+
H'
E
of H'.
norm
everywhere
0
--->
following.
H,'
also,
h almost
Proof.
any
a
and the
4(jxj)
of the
is
is the
2N
2 < q
3 and
addition,
Sketth
H,1.
need
we
scalar
H,' coZnciding
of H,1 in
H.2.4
J.,=0vo
[r(-y(s))]
therefore
(-1,
Then,
0.
[75,76].
from
result
H.2.2
0
obviously
is
,=0
to
>= 0
w
Theorem
Pohozaev
d
1,
7'(s)
kind
-y'(0)
and
vo
=
0.
map -Y from
X(r (vo) vo), -y'(0)
0 there
C
exists
00
h (r)
(s)
lim n-oo
and is
for
any
continuous
a, b on
:
0
r
r
(+oo)
hence
Also,
0.
=
obviously
we
have 2
00
00
1 (r)j
h(s)
ds
I
Rj)
I
r
N-1
(R)jjh,,jjj2,r'
:5 DNCq-2rq-2
Ih.,,(r)lldr
R
Since
here
for
large
ciently
Remark
Sobolev
spaces
Now,
we
functions:
u(jxj)
> 0 and a] I
of functions
from
Clearly
u
(a, b) which
E
r
0
=
0 is
2.1).
Section
see
contained =
in the
particular,
in
0 for
=
0(a)
0, then
>
a
domain
equality
the
wv(x)O(x)
+
If
equation
0
have
we
2
C6b,,P,-'
0 < an
0 be
fUn}n=1,2,3....
sequence
arbitrary
its
set
we
H',
E
u
jV such that
C
fixed;
0 is
>
following.
is the
Let p E
A
JE(u)/
Inf
=
prove
fYn}n=1,2,3,...
sequence
compact
to
A where
=
the
of
fUn(*
sequence
point
is
+
(11.3.1) Yn)}n=1,2,3....
the
of
solution
a
Then L >
problem
the
and
-00
there
exists
relatively
is
problem
minimization
(11.3.1). Remark and any
A
>
If p
11.3.2 0
has
one
+
-I-,N
then
0.
To
see
> I
IA
Remark
providing
the
M N
1 +
e
T
1
(p+l)+N
2
07
P.L.
+0 when p
-4
sufficiently Lions
example,
he
0 and
f (x, u)
11.3.4
As P.L.
relative
is
large
[57,58]
in
c(x)u
p + 1
I I-I--
something Lions
compactness
the
=
like
>
'
x
and,
N
1 u (1, x) jP+1
dx.
for
E
p
(1,
1 +
N
0.
problems
considered
f (x, u),
=
1 +
> cr
investigated
U
c(x)
(1,
RN
-Au +
where
For P E
function
2
N
RN
Hence,
A > 0.
any
the
=A and
(o,, x)) =20r2
E (u
consider
VA
u(c, X) We have
for
-oo
=
of the
essentially
problem N
E R
0,
k(x) ju IP-'u his
with
publications,
noted
in
of any
minimizing
sequence
k(x) the
>
0.
principal
up to
relation
translations
as
58
CHAPTER2.
in Theorem
11.3.1
I,\
is
I,, + I,\-,,
0
c
there
for which
R> 0
U2k(X +Yk)dx>A-c, BR(O) fx (ii) (vanishing) =
k=1,2,3,...
BR +(0)
yl,
(here
satisfying
properties:
(i) (compactness) exists
there
A,
=
RN:
E
jxj
R});
there
exist
[V and satisfying
in
I
u""
(UA;
-
the
U2)1
+
k
(0, A)
fUk1}k=1,2,3....
and sequences
and
JU2k}k=1,2,3,...
following: --+0
k--+oo
as
for
q
I (Uk1)2dX
lim k-oo
G
a
-
a
I (Uk2)2dX
lim
=
k-00
RN
2N
2 0.
RN
With from
Part
the I of
Proof
[57].
of Lemma 11.3.5
We introduce
Qn(0
we,
the
=
actually,
concentration
SUP YERN
f
the
repeat functions
proof of
of Lemma 111. 1
measures
U2 (x)dx.
y+Bt (0)
jQn(t)Jn=1,2,3....
Then, functions quence
that
on
R+
is
and
k-oo
Qn, (t)
=
sequence
liM t-
f Qnk } k=1,2,3.... lim
a
+00
and
Q(t) for
a
Qn(t) function
nondecreasing,
of
A.
=
By
the
nonnegative, classical
Q(t) nonnegative
any t > 0.
result, and
uniformly there
nondecreasing
bounded
exist on
a
subse-
R+ such
OF RLIONS METHOD CONCENTRATION-COMPACTNESS
2.3THE Let
a
+00
t
for
the
Let
us
place
Q(t).
lim
=
Obviously
[0, A].
E
a
fQnk(t)}k=1,2,3,.-
sequence
If
If
a
a
0, then
=
(ii)
vanishing
the
clearly
A, then
=
59
the
takes
(i)
compactness
occurs.
briefly
arbitrary.
be
these
prove
Then,
claims.
two
let
First,
a
0 and
=
let
c
0 and
>
R
0
>
have
we
Qn,(R)
I
sup yERN
=
u',, (x)
dx
exist
sequences
Ck
integers,
such
positive
that
I
claim
+01 Rk
--+
dx
0,
have for
n
+oo
-->
Qn- (Rk)
that
2, (x)
U
proved.
is
k
as
>
Q,,,, (Rk),
=
Second,
k
--+
=
1, 2, 3,
a
all
Then,
A.
=
fMklk=1,2,3
and
oo
for
6k
let
>_
....
...
yj,+BRk(0)
Then, taking
arbitrary
an
>
c
we
I
n_(x Un
k such
all
y,,,)dx
+
A
>
that
mk-:
c.
-
BR, (0)
Now,
to
R > Rk
that
so
the
Consider
place k
in
this
the
case
for
m
all =
k
0,
1,
p(x)
0( ) .
C
>
=-
to
prove
take
to
sequences
Rk
the
that
=
W be
a,
=
A
...
=
(infinitely and O(x)
respectively.
=
exists
Rk,
=
=
k
'Ek,
a
=
k,,,
number
lQnk(4m)
and
...
Rk_+j
get required
< 1
there
m-1
R,
set
we
cut-off
=
m
Q(m)1:5
-
we
Jxj
0 for
and W,,
(iii)
dichotomy 01
Ck >
17
-
El
=
Ck
takes -+
0
as
> 0
Q(4m)l ...
=
such
'Ek,
=
=
that
m-'
k-oo
yk/
+
show that
us
I(a):5
a
ak
=
E (Unk)
lim
=
contradicts
prove
Let
ak
k-oo
Let
a
+
Let
lim
+ E (pkU2)k
IA
A
I(a)
funk I k=1,2,3.... (i) (compactness).
Then
a.
-
E(akUlk)
==
a
property
occur.
=
subsequence
a
the
AI(a)
and, again by is a ball B12 (X2) : B, (X2) C B, (xi) such that p(x) > 2 for any x E B12 (X2) Continue this process. We get a sequence of balls such that f B,. (X,,,)}n=1,2,3.... Bn,, (x +,) C 0 as we can BIn (xn) and p(x) > n for any x E B, (xn); in addition, accept that rn there is > the oo. construction n a n for Then, unique xo E nB,,,(xn). By p(xo) C
-
--+
---+
n>1
any
integer
> 0.
easily
It functional we
n
This
follows
contradiction
from
Indeed,
lemma.0
that
Lemma 11.4.6
in X is continuous.
the
proves
let
admissible
an
Then,
xo E X.
since
lower
p(x)
sernicontinuous
:5
p(x
-
xo) +p(xo),
have
p(x) On the
other
hand,
-
P(xo)
Xx
:5-
-
x0)
M1Ix
0 there
exists
that
P(xo) for
all
x:
jjx
-
xoll
c
that
arbitrary.
p from
show
to
We take
the
number
a
that
P(XO) Choose
It
65
0 such
>
continuity
P(XO)
-
Corollary
PN(Xo) -PN(X)l
that
pN).
of
P(X)
Then,
< PN
11.4.7
PN(XO)
-
(XO)
for
2
x
x
6
+
If.(X)}n=1,2,3....
Let
be
a
xoI I
-
II
xo
-
(X)
SUP Pn
-
2
2
IIx
for
< -
any
0
by
the
of the
convergence
series
en n=O
Clearly,
continuity.
case.
uniformly
M+P
E
0.
STATIONARYPROBLEMS
CHAPTER2.
66
g'= (a,,)-'h'=
Then,
E bkek
n
+ en
0 in H
--*
I
as
But
oo.
--+
E bke-k
then
H as 1
g'
clearly
and
oo
--+
g'
--+
in
kon
kon
bA;ek for
bk,
coefficients
real
some
hence
k96n
1:
+
en
bk ek
0
=
H,
in
kon
i.
get
we
e.
Thus
contradiction.
a
indeed
coefficients
an
linear
continuous
are
func-
in H.
tionals
00
Let
E an(en
F
hn)
-
and
F
Uf.
=
The operator
U is
linear
and
is
it
n=O
everywhere
determined
BR(O)
If
=
E H
11f1l
:
H.
in
R}
0 such
exists
E
that
a
2
M211fJ12
e
exists
number
a
N> 0
large
so
F,
that
I I en
hn
-
a
E
x
solution
as
A, x))
and for
exists
for
[u'(a,
=
all
0 for
increases
2
0 and
>
A > 0 there
any
values
these
whole
>
F(u'(a,
-
(11.4.6),(11.4.7)
n
u(a, A, x)
for
A > 0 satisfies
a
In
integer
time,
A, x)
problem
of the
that
G(A, A)
Au'(a,
+
arbitrary
an
theorem
comparison A.
solution fix
x)]'
all x
[0, 1].
E
x
be
can
con-
Ju(a, A, x) I
satisfies
the
0
--+
equa-
tion
where
the
A > 0 is of the
problem
values
A
A, be the
most
> n
has
0
A
than
more
above
arguments
by
above
arguments.
the in
(0, 1)
as
dependence
a
> 0
has
to the
roots
continuous
of values
set
(11.4.6),(11.4.7)
problem
According
A,(a)
E
(0, 1),
with respect to large uniformly theorem the solution by the comparison
Hence,
(11.4.6),(11.4.7)
x
arbitrary
> 0 is
large.
0,
n
(0, 1)
in
roots
for
E
x
[0, 1]
if
u(a, A, x)
sufficiently
large
> 0.
Let of the
c(A, x)
function
sufficiently
=
function
theorem
such that
at
least
the
set
The
(n
for each of them the +
1)
roots
as
A,, is nonempty.
corresponding
of the
argument u' X (a,
a
A,, (a)
Let
x
of =
x
because
u(a, A, x) E (0, 1).
A,,.
inf
u(a, A.,'(a),
solution
A, xo) :
solution
function
x)
otherwise,
Then has
at
due to the
0, there (a, A, xo) values must exist A < Xn (a) belonging to A,. as a By analogy u (a, A,, (a), x) regarded function of the argument at least in (0, 1) and u(a, A,'(a), x has n roots 0 because 1) in the opposite the solutions, case to A E An sufficiently close to An(a), corresponding must have at most n roots in (0, 1). Let us prove the uniqueness of the above value A A,,(a), for which the solution u(a, An(a), x) of the problem has precisely in the interval n roots (11.4.6),(11.4.7) the condition x E (0, 0. that there exists 1) and satisfies Suppose u(a, An(a), 1) A' 54 A,,(a) these conditions. satisfying Using the autonomy of equation (11.4-6) and and
since
0 if
u
=
=
=
=
its can
invariance
easily
with prove
that
respect
to
the
changes
of variables
x
---+
c
-
x
and
u
--+
-u,
one
ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS
2.4.
(a) for odd with
respect
(b)
the
point
2.
1);
is
the
xo
is
strictly
on
any
(c) u(a, A, x) minimal
ul(x)
exists
the
minimal
value
of the
Ul(X)=h U2(X) byU2(X), the identities
for
x
all
from
2(n+1)
U2(X)
right
a
E
x
(0,
such
x
that
x,
each
that
also
for
point
1
x
f Ul(X)U2(X)
[f(U2(X))
=
where
x,
result
U2(X)
if
2(n+1)
ul(x),
for
the
obtained
[0, Y],
segment
be the
1
=
written
subtracting the
over
Y
let
clear
is
T > 0
or
a
and
it
Let
and
achieves
An(a)
>
0.
(11.4-6),
.
the
and
(11.4.8),
of
view
) Multiplying equation for U2(X), by ul(x),
integrating
to
Section
[0, 1]
E
A'
ul(x)=
that
such
2(n+1)
and
solution;
u(a, An(a), x)
definiteness
the
of the
( 0,
+ 2xi
x
of
respect
also
see
is
u(x)
this
with
increasing
half-neighborhood E
[0, 1];
C
of
2
of two solutions
monotonically
is
'+ 2(n+1) written
equation, another
C
in
1-
"+'2
x0
b]
solution
[XO, X2] and even [0, 1] (on this subject,
Then,
x
+
x0
solution
on
u(a,A',x).
=
b,
this
u.
(a)-(c) (11.4.6),(11.4.7), Let
-
arbitrary
an
extremum
and
any
function
the
argument
same
one
of
and
=
that
2xi) for
properties
point
u(a, An(a), x) ul(x)> U2(X) in
-
+
x
root
problem
of the
[xo
u(a, A,
=
of
point
on
of
< X2
x,
[xi, xo] b, xo + b]
[xo
segment
roots
(11.4.6),
of equation
arbitrary
unique
a
segment
solution
an
nearest
monotone
from the
maximum at
xo
on
two
positive
It follows
u(a, A', x)
point
there
arbitrary
an
arbitrary
(11.4.6)
solution
of
xo
the
to
between
equation this
root
any
69
we
get:
7
0 >
f(U2(X)) 2
_
1
-
An (a) + A'] dx.
(11.4.9)
0
by
since
But
inequality
suppositionUl(X)
our
positive,
is
i.
Lemma 11.4.9
We 11.4.9.
is
keep
Let
we
e.
An(a)
The property
get
(r(n
!
U2(X)
> a
+
for
E
x
(0, Y)
the
,
right-hand
side
of this
contradiction.
1))2
follows
from the
theorem.
comparison
Thus,
proved.0 the
A An(a) for the value of the parameter x)= Un(a, x). By Lemma 11.4.9 these definitions
notation
u,,(a,An(a),
from
Lemma correct
are
1
An (a)
and
0 for
>
any
a
>
0 and
integer
n
Let
> 0.
also
an
f Un' (a, x)
(a)
dx.
0
Lemma 11.4.10 continuous
on
Proof.
contrary, the
U2(X) each
i.
e.
properties increases
of them.
a,
that
>
An (a,)
(a)-(c) on
a2
integer
any
half-line
the
Let
For
0
An(a)
is
nondecreasing
and
> 0.
a
>
n
Let
proof
[0, have
that
prove ul
(x)
=
of Lemma 1-
2(n+1)
u',(xi)
]
and >
x
un
11.4.9, =
U2(X2)
An(al) (a,, x) each
! and
An(a2)U2 (X)
of the
Suppose =
functions
Un
the
(a2 x). By ul(x) and ,
1is the point of maximum of 2(n+1) for any y > 0 for which there exist
(0,
X1) X2 E x
STATIONARYPROBLEMS
CHAPTER2.
70
-(n+jj)-
(0,
E
we
satisfying
2(n+l)
I
I
(xi)
ul
Proceeding
U2
as
(X2)
==
Therefore,
y.
(x)
ul
(11.4.9)
inequality
deriving
when
and
> U2
taking I
(X)
for
all I
-
x
=
2(n+l)
get 2
2(X))
U1(X)U2(X)[f(U
0
f(U2(X)) 2
_
1
n(aj)
-
+
An(a2)]dx,
0
which
obviously
is
proved
is
that
Let that
the
there
the
exists
0 such
>
ao
nondecreasing
is
of the
continuity that
definiteness
the
by analogy). for
each
Then,
1) d(a) 2) u, (a, x)
+0
-+
3)
Un
first
as
one
sufficiently
> ao
a
the
ao+O
as
> un
(a, d(a))
a
inequality
take
easily
verify,
close
to
(ao, x)
E
x
(ao, d(a))
Un
:--:
for
the
An(a). An(ao)
>
(the
place
is
positive.
a
Suppose
the
second
case
0 such
>
it
i.
e.
contrary,
An(ao).
An(a)
liM a-ao-O
d(a)
exists
here
half-line
or
follows
it
there
side
(11.4.8)
and
that
that
0;
+
ao
-
An(a)
can
ao
on
function
liM a
for
right-hand
the
because
An(a)
function
prove
us
contradiction
a
(0, d(a)); -A-
and
< -- - u, dx
(a, d(a))
dx un
(ao, d(a)).
of the point x d(a) Un(a, x) < Un(ao, x) in a right half-neighborhood follows from it Then, as above, equality d(a))). (because 0,xx(a, d(a)) < u",xjao, close to ao and for all that Un(a, x) < Un(ao, x) for all a > ao sufficiently (11.4.8) x E (d(a), ; '-+ ). Using the identity similar to (11.4.9) with the integral over the 2(n+l) a contradiction. we get So, the function ' n(a) is continuous, segment [d(a), 2(n+l) Therefore
=
n
n
and Lemma 11.4.10
an(a)
Lemma 11.4.11 a
+0
a
Proof.-
The
(see (11.4.6),(11.4.7)
Lemma
0
as
a
liM a
+0
(0,
all 2(n+l)
and also
addition,
x
us
in
an(a)
x
G
an
(a)
12[
A.
follows
from the
dependence
continuous a
function
continuous
on
the
half-line
+oo.
=
function
parameters
that
prove
such
2(n+l) that
our
lim a-+oo
1
f(U2)U by
an(a)
lim
and from the the
increasing
Further,
[0, 1] (see
the
as
proof
continuity
of solutions
it
proved
is
of
) n(a)
Of
of the
problem
un(a,
earlier,
10),
Lemma IIA.
x)
--->
therefore
0.
(0,
(-=
strictly
a-+oo
11.4.10)
=
a
of the
continuity
+0 uniformly
an(a)
is
0 and
=
on
--*
Let
for
an(a)
liM
0,
>
proved.0
is
-
).
Indeed,
u"
n,
xx
An(a)u supposition
if
(a, xo) is
a
=
we
> 0.
First
+oo.
f (u 2(a, xo))
all,
we
u',x (a, xo) function
on
n
-
observe
0
it
as
the
An (a)Un (a, xo)
that
then
contrary, >
nondecreasing n
the
suppose
But
of
0.
:!'
0
xo
E
n
there
was
half-line >
u",xx (a, x) exists
indicated u
Hence,
E
earlier
[0, +oo); we
get
in
that
ONBASIS PROPERTIES OF SYSTEMS OF SOLUTIONS
2.4.
Un',.,(a, all
x) E(01
x
0 for
>
all
E
x
(xo, 11,
i.
lim
a,,(a)
e.
have
we
So, un,,(a,
contradiction.
a
71
x)
2(n'+I))'
Now,
to
that
prove
+oo
a
suffices
it
+oo,
=
1
u,,(a,
show that
to
2(n+l)
it is proved as (because, above, Un(a, x) is a concave function on the +oo the following segment [0, n+1 T' ]). Suppose that for a sequence ak C Consider the < two cases: place: :5 +oo. Un(ak7 2(n+l) separately following ; '+- ) < +oo. +oo and B. f (+oo) f (+oo) +oo
as
for
< 0
a
of
+oo
--+
x
takes
-+
A.
=
f (+oo)
A. Let we
would
(11.4.8)
get the
functions
Therefore,
An(a) u,(ak, k-oo
Then
+oo.
=
from
Un
(ak x) satisfy +
n,xx
where
9k
ing
standard
to
+oo
-*
Un(ak7 X)
for
all
B.
:!
an(a)
lim +oo
It
for
all
E
x
that
Un
Then,
(r(n
equation
it
is
We also
note ==
Let
the
1))2
e.
we
that
an(a)
is
x
have
E
(0,
)
the
equations
Hence,
2(n+l)
each
of the
; '+-
(0,
in
root
a
But
+oo.
get
by
hand,
one
this
arguments
in < a2
Suppose
the
Un(a2lx)
for
interval
such
(a2 X) right xj) un,x(a27 Xl)- In An(a2), hence, in view in
7
half-
a
=
Un
An(al) e.
a,,
takes
the
on
0 < a,
:
un(ai,x)
(a,, x)
Un
(11.4.8) written for x xi, we > (11.4.6), u",x, ,(a,, xi) u",xx(a2;X1)i that u,,(al, x) > un(a21 X) in a right .
any
that
such
a2
show that
us
for
x) :5 un(a21 X)
x,
Suppose
a
suffices
To prove
n
proved
; '-+
2(n+l)
theorem,
comparison
on
+
i.
+oo,
E
contradiction).
a
e.
(0,
x
otherwise
01
-=
to
k must
i.
impossible.
equality
of
An(al)
the
X)
respect
numbers
A is
+
prove
; '-+
E (0, (a,, xi)
of
with
on
'-+ ) 2(n+l) e.
neighborhood view
+oo.
=
to
(0,
i. x
xi.
X1)
if
An(a2)X2
X2
(n+l)
when
deriving
be the if there
is
inequality
no
such
(11.4.9)
a
E
point with
(xj, in
the
such
2(n+l)
(xi,
1
2(n+l)
integration
that
Un(aj) x)
). Repeating over
the
the
=
Un(a27 X)
procedure
segment
[XI, X21
or
used ,
we
CHAPTER2.
72
STATIONARYPROBLEMS
get: 272
I
0 >
un(al,
X)[f(U2(a,,
x)u,,(a2,
x))
n
f(U2 (a2 X))
_
An (a,)
-
,
n
An (a2)] dx;
+
X1
addition,
in
An(a2).
by
Thus
increasing
the
above
get
we
function
of the
problem
the
interval
integer
any
the
inequalities
following
Proof.
Suppose x
E I
for
all
x
by
the
[X I
i
2(n+2)
21
X
we
,
as
suffices
following < X2
Un+I(X2)
Un
7
(X)) satisfying
has
precisely
+1
of
in
roots
n
function
the
Un;
It
to
Un+I(X)
strictly
be that
un+,(x)
f (u' (x))
>
An for
-
n
exist
two
with
B
increases
E
x
can
and
also
the
integral
Un(X) hence
over
that
occur.
un+,(x) Un+I(X2)
that
!
I,
E I such
xo
A and
such
Un(X2)7
point
a
cases
(11.4.10).
inequality
prove
cannot
2(n+2)
>
(11.4.10)
...
solution
(11.4.9)
deriving
- An+1 for
that
for
u'
In view
monotonically
a
place:
A2
a
the
is
it
and such that W-4.1)-(H.4.3) (0, 1) and this pair is unique
addition,
in
So,
argument
For
have
we
contradiction.
a
Lemma 11.4.12
the
arguments,
> =
the
un(x) Un(X2)
if if
segment
get the inequality X2
0 >
i
2
Un+l(X)Un(X)[f(U
n
+1
(X))
_
f(U2 (X))
An+1
-
n
A,,]dx,
+
X1
where,
as
Thus,
we
proof
in the
get
a
of Lemma
contradiction,
11.4.11,
and the
the
A is
case
inequality
strict
place
takes
if
An+,
An
==
impossible.
un+-I(x) < u,(x) for all x E I. Observe that Un+I(X) < Un(X) for some 1 have u'( we would (because otherwise 0). Further, un+,(x) :! ' Un(X) 2(n+2) ) is obvious Let x that E [0, on a then us visually (it picture). prove n+1
B. Let xo E I
for
Un+1
the
equal
Y.
k
v(x) Repeat
procedure
this
for
of the
properties
I v (x) I
lies
that
such
k
each
(a)-(c)
under
the
Jv(x)j
Un(bn7 X))
G(, n(bn),
(r(n
!
dn
large
sufficiently
all
we
would
for
all
get that
in
large
sufficiently
So,
contradiction.
X)
n
-
F(u'(bni
X)),
n
(11.4.11)
that
implies
for
1))'u'(bn7
+
the
numbers
(11.4.11)
Indeed,
n.
if
right-hand
the
numbers
< 20
n
and
for
side all
boundedness
uniform
we
suppose
is
than
greater
satisfying
x
of the
this
that
is
the
not
left-hand
20, jUn(bn,x)j jUn(bn7X)Jn=0'1,2.... =
sequence
then
so,
one
i.
e.
a
is
proved. Let
hn(X)
=
properties
us
show
sin[7r(n (a)-(c)
10
that +
from
jun(bni -)IL2(0,1) 1)x] and observe the
proof -h"
> I
that
of Lemma
=
n
/-tnhn7
for
all
u',x(bn) n
11.4.9, X
G
large
sufficiently
0)
=
u'n,,,(bn) (0, 1),
h' n (0)
1)
=
numbers
and,
hn (1). '
in
view
n.
Set
of the
We have:
CHAPTER2.
74
h,(O) where
un
(11.4.6)
(7r (n
=
for
the
+
1))'.
Therefore,
w,,(x)
functions
"+ -Wn
Wn(O) where
family
the
boundedness
x)
Wn(l)
family
C,
constant
a
Multiplying
this
[0, 1]
segment
independent
> 0
of
applying
and the
place
by
the
by parts,
integration
because
12
An I Wn L2(0,1)
_JW/'xJ2
L2 (0,I)
n
-
-
all fact
for
C2
numbers and
all
independent
0 is n,
we
f
n
-)
0, 1, 2,....
=
C3
exists
C3 and
u,,,(bn,
functions
the
for
as
show that
n
2W2(X)dx
X
n.
enIL2(0,1)
shall
for
uniformly
is
-
> I
(11.4.13)
using
Jun
equality
In view of this
n.
1
that
fUn(*)In=0,1,2....
functions
For this
of
the
over
0
JUn(bn, *)IL2(0,1)
have
Lemma IIA.11
sufficiently It
the
>
equality
get
0
where
theorem.
comparison
obtained
we
1 xw',,,(x)Wn(x)dx
2
uniform
estimate
the
1
11.4.5,
Theorem
To prove
function
( n+1 t )
u,,
let
aim,
L2(0, 1)
space
independence
linear
the
this
For the
in
STATIONARYPROBLEMS
for
us
each
inte-
Fourier
the
in
in
series:
00
Ea ne-k(') k
n+1
where
ank
coefficients.
real
are
Then
k=O 00
Un(*)
r
=
b,,,e,,,(-)
(11.4.17)
n
M=0
in the k
=
0, 1, 2,
Then,
=
holds
valid
is
each
and,
n
in view of
real
exist
(n+1,
L2
our
Also,
space
=
a
0
points
r
n+1
the
1,
==
obviously
bn0
because
the
where
n+1
(X),
e(n+l)(k+l)-l
(11.4.17)
equality
too,
where
> 0
n
the
the functions
points,
)
r+1 n+1
r
(01
of Lemma 11.4.9
precisely
are
to these
respect
spaces
proof
the
shows that
verification
for all
,
2,
also
Therefore,
n.
bnn-1 functions
0 for
=
U.
and e.
sign everywhere. that
suppose
us
which
L2(0, 1). acceptation, bnn
of the
sense
same
Let
there
the
of the
from
roots
m: (n+l)(k+l)-i place in the space L2
0 if
=
takes
(a)-(c)
to its
respect odd with
are
obviously
properties
the direct
of each
sense
in
of the
are
is odd with
bn,,
nand k
a
=
(11.4.17)
equality of the
0, 1, 2,...,
in the
bn (n+l)(k+l)-l
where
and since
n
k
,
view
un(x)
where
it
in
1, 2,
1)
Indeed,
....
since
function r
L2 (0)
space
the
coefficients
fUn}n=0,1,2,...
system
Cn,
n
=
0, 1, 2,...,
independent.
linearly
is not
equal
all
not
to
such
zero
Then,
that
00
ECnUn
(11.4.18)
0
=
n=O
in the
b1jcj
L2(0, 1). Let (11.4.18)
space
Multiply =
But
0.
contradiction, in the
L2 (07
1)
large
is
radial
solutions
a
proved.
f(02)
method,
for
Independently, existence
jolp-1 =
in
of
a
contained.
presented;
number
N
In
the
the
existence
Thus,
problem
(p
positive
space
M, =A
0 and
that
co
L2 (0) 1) we
as
of the
Theorem
IIA.5
cl-I
In
-
of
view
supposed
cl
=
:
0.
proved,
is
54
0 and cl
(11.4.17)
we
So,
0.
get:
get
we
a
fUn(X)}n=0,1,2....
of functions
system
1)
>
devoted
publications
of
3 and
[96]
the
too.0
remarks.
of the
=
in
independence
linear is
el
above
Additional
There
is
by
proved
is
and the
space
2.5
with
it
as
I > 0 be such
index
the
equality
is
(11.0.1),(11.0.3).
for
the
solution,
[97]
paper
of
based
a
a
solution
f,
function
same
on
methods
refinement with
of
an
I
[71],
a
< p
of the
of the
arbitrary
[71],[96],
papers
paper
existence
of the
questions
In the
In the
considered.
1 < p < 4 the
the
to
< 3
and
N
qualitative
technique given
a
solution =
theory of the
number
variational
proved.
is
3
of
problem
this
by using
positive
existence
proof
a
of
paper
of roots
of
ODEs,
[71] on
is the
77
ADDITIONAL REMARKS.
2.5.
half-line
r
[82]
in
[71] (N
the
the
there
in
bounded
solution
is
i.
satisfies
it
e.
also
applicable
f (0') 0
in
principal achieving
the
W. Strauss
[87]; (non-radial)
mistake
no
nontrivial
=
paper
solutions
that
stating
Till the
101 investigated has
0; sufficient
a
--+
have
reviewed,
Another recent
paper
on
root, for
conditions of roots
case
positive
the
the
half-line
solved
of
is based
a
similar
to are
nonlinearities
for
proved
first
[98];
in
of solutions
of the existence
unsolved
remained
has
aim but
for this
exploited
method.
variational
obtained
by
by
methods
of the
a
the
(11.1.4),(11.1.5)
Cauchy problem
was
theory
qualitative
of
model
is considered
[110]. w-f (0)
> 0
r
of
jxj
an
it
paper,
is
radial
solution
are
obtained
101-1,
=
11.1.2.
supposed with
a
of
results
the
case
f( 02)
One
more
the
an
by using
made
number
that Jim i(Al-00
finite
a
so.
in the
a
of
has
given is not
(02)
Theorem
exists
> 0
f
it
(11-0.1),(11.0.3)
and there a
author
arbitrary
[77,78];
proposed
was
symmetrization
a
the
example
with
In this
=
with
problem
existence
concept
Rabinowitz
the
of the
of
Unfortunately,
solution
a
of P.H. with
the
solutions
radial
positive
on
H1.
from
solutions
oo.
0
>
existence
from results
in the
unique
the
method
existence
of radial
existence
as
number
we
was
of the
r
any
[47].
function
the
directly
now,
half-line
for
two papers
3 and was
(3,5),
p E
ODEapproach
an
completely
proving
jolp-1,
of solutions
the
problem
the
=
:
11.1.2
also
[110].
from
f(02)
3 and
see
his
101P-10.
N
Theorem
latter
0,
=
possesses
A result
0.
of the
r
paper
101P`
=
>
r
for
solutions
methods
[10],
was
[59].
paper
of radial =
half-line
Methods
positive
that
result
f(o')
3 and
=
the
on
ODEapproach,
=
on
N
of this
the
are
solved
point
of the
have
we
an was
solutions
I < P < 5 any
methods
fact,
proof
completely
was
for
that
that
so
with
considered
neighborhood
a
OIP-'
f (0')
3 and
(11.0.1),(11.0.3)
problem
In
of roots
existence
in
follows
roots
N
Shekhter,
way of
nonnegative
g(o)
p
4 < p < 5 the
for
This
in
with
f(02)
paper
Another
+oo
proved
is
5 in the
for
=
problem but
paper
sign
in the
of the
value
B.L.
this
N
the
(11.0.1),(11.0.3).
exploited
problem
by
of ODEs
in
alternative
to
the
on zero
The indicated
on
are
whether 0.
=
derivative
number
we
In the
time.
r
(11.0.1),(11.0.3)
with
result
first
its
sense
a
(11.0.1),(11.0.3) long
proved
been
the
framework
In the
by
there
for
5,
0
we
the
on
is
Also,
[119]
in
way consists
that
in
exploited in
upper
of a
general
triangular
even
for
the
in
basis
a
analog
attempt
[118].
to
properties
and
all
system
11.4.5 use
the we
of the
elements of functions
of its
on
expansions note
that
where
JUn}n=0,1,2....
of
s
that
the
< so
and
on
the
is considered.
Bary
the
(11.4.17) b,',
(the
diagonal to
theorem. for
example
an
coefficients
principal
probproblem
proved
problem
based
which,
transform
Fourier
eigenvalue
However,
the
in
of the
of Theorem
proof an
is
an
is
systems
boundary-value
[1181 H'(0, 1)
IIA.5
Similar their
is
of
monograph errors
for
first
discussion
[119].
it
paper
nonlin-
in the
was
essential in
a
of the
small
of this
basis
a
[118],
In
nonlinear
a
being
In the
denumerable,
eigenfunctions presented is
property
results
interesting
of Theorem
proof
published
knowledge We only
thorough
approach
the
eigen-
completeness
the
contains
of
direction.
some
more
The first
of
page
111.
best
arising Bary theorem
the
are
that
under
A
proof
the
is considered.
which
approach
[39];
[6].
in
[115-117].
in
[6].
in
in
paper
The
corrections
presented
are
proved
contained
in that
parameter)
over
and
approach
the
results
L2
have
shows
[5]
proved.
is
These
constant.
is (bn )n,m=0,1,2,... insufficient are zero) L2(0) 1)M
from
solutions,
natural
This paper
in
However,
spectral
negative
half-line
aim.
in
of its a
proof)
corresponding
eigenfunctions system
a
is
equation,
operator
problem,
theorem
this
in
note
system
author's
[62] containing author [63] where
same
linear
be corrected.
can
and
lems
[114].
in
the
nonlinear
a
results
no
on
of the
papers
wo- 101"o,
=--
error
properties to the
nonlinearity in the
We also
principal
a
uniqueness
the
(for g(o)
exists).
problem,
almost
there
of positive
proved
is
proved
it
basis
on
for
Makhmudov
by
paper
differs
theorem
2.4
are
a
this
around
questions this
by
(if
contains
Liouville-type there
A.P.
of
of
(without
announced
to
and
and the
perturbations
ear
field
eigenelements
of
[83]
in
as
unique
from Section
Sturm-
monograph
subject
this
on
new
a
the
mention
presented
the result
nonlinear
a
quite
is
always
is
particular,
it is
the
uniqueness
solution
positive
[54]:
in
knowledge
our
concerns
the
on
In
of the
of
chapter)
results
(11.0.1),(11.0.3).
is obtained
solution
only
best
introduced
proof.
this
in
are
uniqueness
result
result
similar
Concerning this
the
-
of
there
problem
of the
solutions
considered
literature,
In the
==
(not
problem
The second
methods
to the
the
to
of the
variant
our
close
are
However,
precisely
containing
paper
no
11.2.1
Theorem
proving
our
be
this from
matrix are
complete
non-
in
Chapter
3
Stability
of solutions
it
chapter,
this
In
noted
is
Sobolev
JxJ
with
respect
the
to
distance
stability
the
study
to
of functions
spaces
the
p in
solitary
of
we
(for
definitions
named
a
solitary
see
u(x, t),
wave
Introduction
where
(x, t)
u
O(w,
=
=
x
x
mathematical
pioneer
by
paper
field.
investigations
in the
like
vanishing
solutions
respect like
the
to
O(x -Lot) can
or
such
that
the
i.
the
"forms
if
Thus, the close
two sufficient
which
we
u(x
sense
of the for
conditions
suppose
t)
-r,
graphs"
as
to
x
be
--+
solution
u
p for
O(x
-
graph
wt)
the oo
of
stability for
one-dimensional,
P.E. Zhidkov: LNM 1756, pp. 79 - 104, 2001 © Springer-Verlag Berlin Heidelberg 2001
our
of usual
as
is
of
is
an
arbitrary
perturbed
---->
with
i.
respect
e.
stable,
to
the
Simultaneously
with
N
=
1.
wave
of functions 'T
"almost
then
for
solution
distance we
close
functions
these
almost
are
In Section
coincide".
KdVE.
spaces
u(x, t)
as
and
clearly
x
soliton-
a
travelling
translation
a
functions
O(x -wt) oo JxJ
p "almost
distance
0, then
of
with
u
=
KdVE are
standard
of the
t >
t > 0 there
of the functions
vanishing
visually:
of distances
for this
and
(x, t)
some
of soliton-
form a
the
numerous
f (u)
if
point
a
waves,
stability
the
of
stability
the
a
0(+oo).
further
the
KdVE
is
xo
=
KdVE with
understood
easily
sense
however,
and of the
graph
a
in the
solution
vanishing
solutions
other
of
of the
in the
to it
stability
be
distance
of the
spaces,
graphs
of its
forms
this
and
oo
-+
each
soliton-like
a
x
sense
Sobolev
Lebesgue e.
to
of the
can
proved
has
standard
oo
p; he called
as
in the
close
be not
x
origin author
the
the
that
E R and
x
0,
=
initiated
which
d for
recall
of solitary
stability
the
to
the
was
paper,
--->
terminology
vanishing other
to each
In this as
distance
This
solution.
[7]
Benjamin
T.B.
devoted
literature
as
of the
all
x
In the
vanishing to
case
-
are or
We also
Section
or
NLSE
Lebesgue
KdVE and
of the
case
as
waves
3.1). wt) in the and u(x, t) e'wto(w, x) for the NLSE, a kink if 0' (w, x) = 0 for if there is a unique xO E R such that solution soliton-like 0' (LO, xo) and the of x function of a of extremurn 0(-oo) argument O(Lo, x) as
NLSE
of p and d
As
waves.
KdVE and
of the
waves
of standard
and it is natural
spaces oo
-4
respect
solitary
usually
distances
to
of solitary
stability
of the
questions
consider
Introduction,
the
in
with
unstable
shall
we
3.1
as
r(t)
=
G R
identical, coincide". each
t
> 0
sufficiently we
consider
p of soliton-like
study
the
NLSE
STABILITY OF SOLUTIONS
CHAPTER3.
80
As it behavior
noted
is
as
x
Section
3.2
p under
assumptions
we
x
In two
stability
cases
prove
solitary
of
stability
of
stability
a
the
I
can
=
and kinks. the
to
respect
In
distance
defining
NLSE nonvanishing
interesting
new
vanishing
solutions
stability
the
of the
waves a
type.
solutions
of soliton-like
with
N
on
type.
stability
We begin
considered. one-dimensional
NLSE with
KdVE with
of soliton-like
the
section,
In this
assumptions solutions
soliton-like
are
of the
of kinks
a
we
Stability
3.1
these
natural
and the
of solutions
x
types:
consider
we
in
two
general
of
3.3
oo.
--+
only
the
prove
In Section as
of
waves
KdVE under
the
2.1,
oo of derivatives
--+
solitary
have
Section
in
of such
as
x
solutions
for
be
will
00
-+
KdVE and
the
NLSE. Let
p(u, v)
=
I Ju(-)
inf TER
v(-
-
r)
-
H1
E
v
u,
and
d(u, v) where
H1 is the real
prove
that First.
we
in the
space
in each
case
I Ju(-)
inf yE[0,21r]
'rER,
first
greatest
the
Cauchy problem +
ut
lower
f (u)ux
remark
for
We also
u(.,t)
let
[0, a) I Ju(-, t) 111
continued
solution >
Let
C for
1), (111.
1.
=
1.
2)
C,
0 be
in
t C-
in
I,
the
(uo
> 0
then
[0,
a
interval
an a
twice
a
constant,
existing
0,
2 E H
(x);
(111.1.2)
H2-solution
from Theorem
holds
1.1.3
continuously
differentiable
differentiable
function
twice
(M.1-1),(X.1-2) if
exists
a
=
0).
that
exists >
in
If there
6 > 0 such
8) (resp. there of time [0, 6), 6
+
1.
R,
x, t C:
continuously
twice
0, if
in
in view
an
interval
case
of the
a
=
solution
the
H2 -solution
a
=
can
be
embedding
> 0
of
time
that
u(-, t)
can
be
of the problem
0). u(., t) be a H2_ [0, a) or I [0, a], considered by analogy).
function of time
0
C
and
such
interval
an
exists
a
differentiable
continuously
1.
easily
can
here.
problem
there
a
One
second.
KdVE:
the
arbitrary
an
(Ill. 1), (111. 2) < C (the IV: Jju(-,t)jjj
problem a
of a
interval
f (-) be
0, bounded >
all
the
Let
of the
[0, a],
be
uo
of
result
f(.)
Let
H'-solution
I
onto
Proof.
a
a
or
be
-
taken
uniqueness b
exists
H2-solution
a
f (.)
for
obviously,
Then
due to the
there
fl(.)
C and let
u(x, t)
solution
t > 0 and
half-line
entire
[2,4);
the
NLSE:
0,
X,t
E
Then,
0 such
half-line
and
uo
of
this
H'
E
0 and
>
formulating 111. 1. 1 if
in
of the
>
according one
a
proves
suitable
parameter
u(x, t)
solution
method
following to which or
result
E
f (.)
supposes
sense
of
We consider on
connected
2 is
v
satisfy-
the problem for all t > 0
solutions. the
v
V(x,t)
solution
u(x, t) of t
of soliton-like
However, values
any
uniqueness
Proposition
(111.1.1),(111.1.2)
this
concentration-compactness
stability When
call
for
H'-solution
entire
existence
we
that
requirement
other
(111.1.3) soliton-like
=
H'.
of the
0.
R,
f(JU12)U in equation e'wtO(x) be a U(x, t)
let
the
function.
problem
b
uo(x).
the
onto
of the
Jul',
the
for
corresponding
application
investigation f (u)
differentiable
continuously
c (the L2.4).
an
of Theorem
sumptions
0 there
corresponding
the onto
of
solution
local
unique
a
t > 0.
iUt + AU +
Definition
then
Cauchy problem
the
if for
be continued
can
all
has
soliton-like
a
stable
O(w, .))
H and
the
one
w
solution
2
(111.1.1),(111.1.2)
problem
the
wt),
-
call
we
H2
E
uo
0(w,
let
Then
if
that,
such
of
for
111.1.1
sition
with is
a
the as-
twice
the
(for example, (0, 4), then all
local in the
the
from
arguments
Theorem soliton-like
Let
O(x,t) from ftn}n=1,2,3.... as
n
111.1.4
is
(n) uo
=
Let
the
=
has
solution
a
clearly
and
boundary
> 0 and
w
E(u)
functional
is defined
I V
(the parameter
w
this
boundary-value
belongs
01
two functions
A,
we
problem n --+
n
JUn(*) tn) 122 a
02
the
P(U(n), 01t=0)
0
_
0
Cauchy
of the
Theorem
to
following
for a
A,
(11. 0 (up
family =
1
-3)
11.3.1, with
equation
is
shown
it
some
2.1,
of Section
2.1, 0 that
solution
positive
of the
values
of
y,,)
any
according minimization
our
+
for
parameter
Therefore,
wn(-
that
102 122
10, 122 :
since
different
C R such
in Section
beginning
translation). fWn}n=1,2,3....
a
sequence
fYn}n=1,2,3....
sequence
it
the
Further,
with
to
0
translations,
0.
>
=
as
As at
up to
minimizing
any
U(00)
solutions). some
A and
=
our
we
take
--+
A
as
vn n
--+
=
oo
-+
e.
in the on
the
a
(-,tn) .Since JU.(*,tn)12 and E(Un('7 0)) Aun
for
sequence
n
-+
contradiction.
111-1.5
For the
papers
by
stability
the
above
C R such as
fUn}n=1,2,3,...
of solutions
sequence
jYrjn=1,2,3.... 0 p(vn, 0)
oo7 i.
fact,
results
sequence
a
1.1.3.
According
2.3.
otherwise,
unique,
a
with
from the
sequence
--+
0,
>
E(u),
LOU,
=
because
has
(11.1.3)
and
to
Remark In
problem
exists
minimizing
Hence, n
solution
0
--+
in H'
as
oo.
any
a
nontrivial
11.3.1,
there
Return For
positive
no
A,
that
get
Theorem
to
be
has
family
the
to
jul"u
+ 1
must
problem
above
is stable.
problem
satisfies
solution
its
every
E
that
H2-solutions
are
3)
1.
conditions
U" +
the
in Section
(11.
exist
by Theorem
given
inf JU12=A>o 2
uEHI,
the
Un(Xi t)
A > 0 the
any
corresponding
the
there
minimization
IA where
that
from H2such
where
e
for
family
the
Then,
0
P(Un(*i tn)i Olt=tn) > with uo (111.1.1),(111.1.2) lo(.,t)12 2 A > 0. Consider
problem
Suppose
ju(n)}n=1,2,3....
sequence
a
Then,
1) from
1.
stable.
not
[2,4).
E
v
(Iff.
A > 0.
(11.1.3)
R+ and
C
arbitrary
an
where
Kd VE
the
hold.
below
Jul"
=
and
oo
-+
fix
family
the
f(u)
Let
O(x, t) of
us
theorem
of the
proof
solution
Proof.
is
STABILITY OF SOLUTIONS
CHAPTER3.
82
with
proofs
+
lun (-,tn)
12
time,
based
on
1,
111.1.4
Lions the
is
[23]
we
1.1.3
is clear
it
that
therefore in
converges -->
111.1.4
Theorem and P.L.
Theorem
problem,
y.,,)
A
Theorem
Thus, first
tn))j
v,,(.
that
T. Cazenave
E(un(-,
minimization
Since
oo.
according
to
=
problem
of the
get
H' to 0 P (Un
('I tn)
JUn(.,0)12
2
f vn},,=,
2,3....
there as
exists
n
oo.
--+
0
7
as
proved.0 was
and
concentration-
proved by
P.L.
in the
paper
Lions
compactness
[57,58]
[1011. the
method
83
STABILITY OF SOLITON-LIKE SOLUTIONS
3.1.
for
investigated
in these
applying
this
method
Now we consider
the
"Q- criterion"
possibility
ishing
of
jxj
as
"Q-criterion"
name
often
by
P and
stability of the
KdVE
in the
the
close
used
waves.
to
conser-
of the
condition
in the
law Eo
conservation
the
van-
necessary.
the
in which
here
rename
variable,
example
solutions
stability
KdVE)
We also
a
is
solitary
of
literature
physical
of the
case
by Q.
denoted
been
has
(E0
NLSE
law P of the
vation
originates
the
an
of soliton-like
of the
conditions
spatial
with
stability
of the
stability
of the
from
on
particular,
in
the
illustrate
to
problem
to the
sufficient
gives
which
oo
--+
only
wanted
we
considered:
are
depending
coefficients
Here
papers.
kind
general
more
NLSE, admitting
multidimensional
The
essentially
NLSE of the
a
U
that,
recall
the
in
7(u)
KdVE,
the
on
case
=
f f (p)dp
and
0 U
F(u)
f 7(p)dp.
=
0
Theorem
111.1.6
plex argument
f(JU12 )u NLSE (111.
Let
for the
u
differentiable
continuously
be
a
1.
3)
with
(f (u)
N= I
of the
function
differ-
continuously
be twice
com-
'r
entiable
for
KdVE).
the
f f (s)ds
F(r)
also
Let
and let
there
wob 2
0 and F (02)
exist
wo E
R and b > 0
0
f (0)
that
such
-
0, f (b 2)
wo
L,,o
-
wo < 0, 7(b) (0, b) (resp., f (0) the KdVE). As for 0 E (0, b) for
0
G
there
tions
U(x, t) 0(w,
exists
O(wo,
x
E
-
wot) for
it
-
0, F(b)
>
proved
is
U(x, t)
solution
soliton-like
a
wob
-
-
F(b 2)
0,
2
!Ib
-
2
e'woto(wo,
=
as
x) for
these
0
In
'.
A
N
[15].
in
v(x, t)
the
in the
NLSE if p
We
(111. 1.3).
prove
the
if p E
proved
of variables
that
0 if
By analogy,
NLSE.
NLSE
2,
>
from
results
0 is satisfied
>
place
the
can
and
condition.
dw
consider
of the
12 ]dx.
this
takes
the
1)
=
dp(o)
for
> 4
making
12 +wolh(x)
under
instability
solution
a
N
condition
the
We first
wo > 0
u(x, t)
where
is stable
when p
111.1.6.
(for
111.1.6
the
instability
of Theorem
11hl 121
=
show that
and, respectively,
0
=
if necessary, that
we
U(x, t) I u IP
f (u)
KdVE with
follows,
similar
f (0)
According 1) the solution
Theorem
to
Chapter dw
RN
0 < p < -1. N
solu-
=
also
accept
lower
bound
we
greatest
that accept 6- f (O)tu(x, t)
-00
in the
expression
remark u
that
in the
real).
d( U, u)
for
generally
is achieved
and
-r
e-'(-/W+w0t)u(-
form
Differentiating
7
t)
r(t),
+
the
expression
T(t)
7(t)
(we a perturbed solution unique). 0 + h(x, t) where h(x, t) v + iw (v, w are with to and T d(U, u) respect 7, we get at
some
T
=
E R and
E R
We represent
not
are
=
for
=
00
I
V[f(02)
+
202f/(02)]
O'dx
=
X
(111.1.5)
0,
00
00
j
Wof(02 )dx
0.
=
(111.1-6)
00
Further, AE+
WO
dx!
AP
a(s)
where d2
2
=
o(s) f(02).
=
+ W0
_
as
Lemma 111. 1.8
s
--
There
2
[P(O+h)-P(O)]
+0 and
C
exists
+
dX2
0 such
1f(L+v,v)+(L_w,w)}+a(jjhj
12)
2 d2
L+
>
>
that
W 0
(L-
-
w,
[f(02)
w)
+202f/(02)]
CIIWI12 for
all
WE
(111.1.6).
satisfying Proof. of the
Wo
E(u)-E(O)+
operator
Let
w
L_
=
ao
+ wi-
where
corresponding
(L-w,w)
to
=
(0, wi-) the
=
0.
eigenvalue
Then, A,
=
since
0
0,
have
(L_w.L,w_j -) ! A2 JU,_L12 21
we
is the
eigenfunction
H'
STABILITY OF SOLITON-LIKE SOLUTIONS
3.1.
because,
and,
hence,
is
[28]).
see
0
since
is
I
(on
L-
operator
0 is minimal
=
this
subject,
+IW_L0f(02)dX
02f(02 )dx
0.
=
(012
W002 )dx
+
0,
>
get:
we
-00
-00
Jal IW12
C11WI12,
!5
and of
independent
C2
>
sup
I f(02) 1)
0
A,
00
00
f 02f(02 )dx=f
hence
1 2 is positive
L-;
eigenvalue
of the
spectrum
of
spectrum
by (111.1.6):
Then,
a
Since
positive
corresponding
the
of the
point
isolated
an
of the
function,
positive
a
bound
lower
2 is the greatest
where
85
(111-1.8)
therefore
H'.
E
w
CIW112i
:5
(L-w,w)
implies
k
For
independent
0
>
C21 W2 12 with
! of
some
(M
have
w we
X
(L-
I
w)
w,
(1W 1122 )k -+1 + 1
1
W01W122)
+
2f(02 )dx
W
00
Thus,
(L-w, w) ! for k
k >
>
sufficiently
0
0 the
I
00
1
small
independent
and
k+1
2
i-T-1 OW/12
+
W01W12). 2
IIWI12 1
k + 1
of w, because
f W2f(02 )dx
(IW112+WOIW12)_ 2 2
1
expression
k
k
+
for
is not
a
sufficiently
smaller
small
than
-00
00
C2 k + 1
IW12
I
k
2 -
k + I
W2f(02 )dx
(C2 k +
>
k _
k + 1
1
M) IW12
>
2
0.
-
-00
Lemma 111.1.8
In what
is
proved.0
follows,
condition
the
we use
(V, 0) Lemma 111.1.9
(111.1.5)
satisfying
Since
Proof. value
there this
There
2
=
0 and
exists
eigenvalue
an
(111.1.9).
clearly
0'
0'
has
is
an
precisely
eigenfunction is
C
exists
and
minimal.
Lot
gi
that
eigenfunction > >
root,
of
92
(L+v, v) >CIIVI12
L+
A2 is the
0 with
0,
(111.1.9)
0.
such
> 0
one
g1(x)
=
a =
with
second
corresponding
mo'
be
the
for
all
corresponding of L+
eigenvalue
eigenvalue
eigenfunctions
A,
of the
E
v
H'
eigenso
< 0
that
and
operator
CHAPTER3.
86
L+
o,',
normalized =
W
ag,
L2 and let
in
b92
+
01,
+
v
192
+
+
(L+v, v) It
follows
from
the
spectral
C,
of
v)
0
Using
now
Schwartz
the
using
0
v-L,
0-
this
obtain: 00
JIMI
j
00
(of 12 f(02) X
+
202f/(02)
}dx
j
11m I
=
00
[(0/1
XX
WO(of )2]dX X
-00
CC)
1 0/ [f(02) X
+
202f/(02)]
(kgl
+ vi
-)dx
where
VI
=
(L+v where
subspace
L_L be the
kgl
=
STABILITY OF SOLUTIONS
with
C5
>
0.
(111.
1.
14) implies:
v-L)
=
(111.1.14) equality,
we
STABILITY OF SOLITON-LIKE SOLUTIONS
3.1.
Proceeding we
further
proof of
end of the
at the
as
and Lemma 111.1.9
CIIvII1,
::
Lemma III.1.10
I lh(.,
Proof.
for
We have
IIU(',tl)
t) Ill
is
e
-
i('Y(t2)+WOt2)
I I h(.,t2)
0(.
Now h
Theorem
prove
Then,
=
by (111.1.7)
111.1.6.
P(O
+
aj(0)I
-
lim
0. 8
+0
AE +
WO
2
all functions condition
Also,
I I,,
JIU(*itl)
h
Let
P(0)
-
-
ao
=
=
-(t2))Ill
T
C2(lllmhj-+a20l, 2 where
,
-
-u(*,tl) :5
_
of t; hence
lai(t) Further,
ei(Y(t2)+WOt2)0(.
7-(t2))Ill
functional
the
AP
independent
1
+
proved.El
is
we can
0.
7
1 IU(*,t2)
:5
_
U(* t2)1 11
-
Ilh(-,t2)III
-
t.
e'(^f(t2)+WOt2)0(.
-
JIU(')t2)
-
I IU(i t1) I 1,
of
function
JIU(*)t2)
-
7(t2))Ill
-
III -I Ih(-,tj)
and Lemma III.1.10
0,
>
IIh(-,t2)II1
-
7(tl))Ill
-
I Ilh(.,tl)lll
the
inequality
and t2:
tj
O,
jail
:
For
large
I IIrn
+
h
6 > 0 let
+
,
t)
Suppose
+
IlRe h_LIIj
Let
us
prove
a20(',
+
h(.,O)EO6
06 be the
a20111
constant.
h
STABILITY OF SOLUTIONS
t)II1
+
neighbor-
open
0 and with u.,&, t) problem (111-1.3),(111.1.4) sequence hn(', 0) E 06,,, either +0 as n 8,n c or I IIrn hi -n( -, tn)+a2n(tn)O(*7tn oo, such that aln(tn) )112+ 1 2 > h c2for First of some n > IlRe In tn tn) 112 0, 1, 2,3, all, (111. 1.15) implies that for all sufficiently < euntil large n wehave laln(t)l IIImhj_n(*it)+a2n(t)0 t) Ill 2+ E and t IlRe hj-n(., t)112 I < _C2f2 because if lain(t)l + a2n(t)O ( IlIm hj-n(.,t) 1112 + -6 V < h IlRe E2, then by (111.1-15) n(', t) 112I ! 2 +C562 which is a contradiction lain(t)l as
+0.
-->
this
right.
is not
of solutions
a
-->
Then,
there
exist
a
of the
-+
....
=
_
-
1
-
because
tn
>
0,
n
Then,
0
c
large
all
(111. 1.16),
to
independent
sufficiently
[AE
of n,
large Wo
+
Thus,
we
arrive
which
the
solution
2 at
taking
we
numbers
+
to
the
place
for
all
exists.
soliton-like
for those
t > 0 for
solutions solution
which
exist
sufficiently
a
Loo'2 'P] Ltn At the
[AE
+
I I Re h_Ln ('; tn) 112
+
IlRe hj-n(*,tn)lll];
large
2
=
02,E2.
>
m-'
and
sufficiently
a
solutions
AP] I
t=0
:5
also
c8i
t >
exist.
t
C7,8,n2
=
yield
1.
0 in the
Hence,
0
-+
as
a
n
--+
proved
are
priori
get
we
oo.
for
all
t for
estimates
0)
(111.
problem
point
(111.1.17)
from
(111-1-18)
relations
0
time,
relations
of the
the
C6
>
same
wo
and
These
u(x, t) V(-, t) at
these
+
n:
-
contradiction,
u(., t)
there
get
n.
API I t*=tn a
assumption,
our
+a2n(tn)0(',tn)II1
JIU(*it)lll taking
112
numbers
m[IIIm hln(*,tn)
[AE for
to
that
sufficiently
all
where the
0
s
-->
NLSE,
of the
case
as
+0 and L we
=
-4 -
-
dX2
+
h E H1
all
[P(u(.,
t))
-
flo(wo,
.))]
!
Loo
f (O(wo, -)).
-
Proceeding
further
as
in
get the estimate
(111.1.19)
satisfying
2
7(1 IhI 11),
+
(Lh,h) for
Wo
El (O(wo, .)) +
-
>
the
and
C911hl 121 condition
00
j
h (x)
0 (Loo, x)
dx
0.
=
00
The end of the
111.1.6
is
proof
Under
III.1.11
NLSE and
KdVE the
d
In
< 0.
the
only these
the
our
of the Here
restrictions,
N,
2
(to
then
we we
2, the problem
< P
a
the
0 for
radial
f(JU12) w
>
N
IUI,'.
0 has
positive
for
a
1, 2);
cases
takes
[15];
paper
=
=
in both
distances
only
questions
case
(11.0.1),(11.0.3)
111.1.6
above
for
the
in
the
to
solution
addition,
solution
place
NLSE
According positive
0.
of the
if
NLSE,
and,
to
results if and
if p satisfies
According
to
CHAPTER3.
90
111.1.7,
Remark
the a
Let w
wo > 0 and
positive
a
the
prove
)
4
N-2
positive
radial
and
b such
constants
all
x
a
e"OtO(wo, x) a
complex
Section
in
as
of the
(p
4
>
N
for
with
respect
to
We present
N=
only
solution
h
=
ih2
+
for
given
a
(11-0-1),(11.0.3)
problem
of the
IVO(x)l
+
Further,
h,
0
all,
of
First
1.3.
number
there
exist
that
z(t)
=-
0 for
can
easily
the
jY0+h(0)-Y95(0)j
arbitrary
not
and
functions
Cauchy problem h
is
function
these
of the
function
a
(1.2.19)
of
decreasing
t)
+ h (x,
choose
can
of the
values
are
2. 1:
Conditions for
a
kink
O(w,
x
providing -
the
wt) satisfying
existence
of kinks
equation
(11.
1.
1)
are
and
STABILITY OF KINKS FOR THE KDVE
3.2.
0(oo)
conditions
the
ing conditions
0
=
it
sufficient
is
and
f f(s)ds,
=
that
necessary
.0
(hereAo)
satisfied
are
exist,
to
91
fj(0)
7(0)
=
-
the
follow-
wO + wo-
and
00
f fi(s)ds):
F1(0)
0-
A:
fi (0-)
B:
F, (0-)
C: F, We also
fl (o+)
0;
F, (o+)
(0)
for
< 0
0;
all
(0-, 0+).
G
require -W
Clearly,
(111.2.1)
condition
provides
I O(W, X) Without
(111.2. 1)
and a
suitable
the we
result
loss
shall
of
(111. 1. 1),(Ill. following.
Theorem
Then,
.
there
of
these
with
Let
the
such
solutions
the
that
of the
of
a
0-
> x
-
0-
Under
wt).
solution
u(00,
infinity
the
>
i
O(w,
kink
on
C1 C2
0+
that
uniqueness
conditions
For this
u(x, t) t)
aim
A-C we
of the
This
need
Cauchy result
(111.2. 1) be valid, f (-) be a twice and a function uo(-) be such that u0(-) O(w, -) E and solution the a (0, a) unique u(x, t) of problem For any O(w, -) E CQ0, a); H2) n C1([0, a); H-'). A-C and
assumptions
-
half-interval
a
1XI)
!5 C1 e-C2
accept
conditions
function
exist
(111.1.2)
stability and
(111.2.1)
< 0.
estimates
we
existence
differentiable
continuously H2
generality
1.2)
111.2.1
the
A0)
I Ox'(W, X) 1
+
show the the
on
problem is the
0 I
-
+
u(-, t)
-
quantity
2IU (X, t) 2
IM-1 0)
=
F, (u (x, t))
-
x
dx
00
does the
that
of
not
above
depend
I Ju(-, t) this
-
solution
and
exists
0(w, -)111
The Proof 1.1.3
i.
t,
on
solution
onto
a
of this
Proposition
there >
law.
0, and there exists
a
In
exists
(unique)
addition, continuation
0.
by analogy
with
the
proof
if
C > 0 such
of Theorem
STABILITY OF SOLUTIONS
CHAPTER3.
92
Remark
111.2.1,
111.2.2
suffices
it
analogous
nition
Remark
t))
get
write
to
I.I.I.
careful
a
the
Since
111.2.3
I(u(-,
quantity
To
to
equation
by
IF 1 (U)I
construction
is well-defined.
difference
the
for
A formal
u(x, t)
of solutions
definition
u
C(U
0 such
that
if uo(.)
corresponding can
Pq(U(',
t), O(W, -))
We first
111.2.4.
_
0
place.
takes
E
-
the
to
respect
u(., t) of
solution
be continued
that
the
stability
from
prove
the
following
estimate:
CP2(U, 0)
I(0)
q
+0 and C
-*
T
=
T(t)
>
Theorem
111.2.4
(P2(U' 0)),
-a
is
chosen
(111.2.2)
.
independent
0 is
C- R is
for
of
Let
u.
Pq(U, 0)
to
u(x, t)
=
be minimal
O(W, (one
x
-
can
T). Then,
of such
AI 00
=
I(U)
-
1(0)
00
Ih,2
2
kink
form.
J(U) where
corresponding
qlu(x)
+
111.2.1
see
conditions
Let also
set
exists
the
easily
can
be the
T)122
_
Then,
0 there
then
inequality
any t > 0 the
1}.
+
>
c
0
technical; 0- < 0+.
is that
minimizing
number
have
a
with
X_+00
definiteness Let
c
=
(a)
assumption the
for
of
0-]e-"
-
+
-
1, conditions
(111.1.3)
NLSE
existence
U02 -i
_
STABILITY OF SOLUTIONS
-
X_+00
follows.
H1.
-
We also
We denote
(since 0(0(-) E Hl).
-
0'(x)e'
lim
=
T)
-
suppose
by
0(-)
-
Of course,
0.
=
a
To
real
E
H',
as
earlier,
we
non-unique.
X'-solution
u(x, t) of the NLSE (111.1.3) be such that lu(-, t) I 0(.) E H' for some t > 0. We set v(x, t) for g(x) u(x To, t), where To is taken lu(x, t) 1, and a(x,t) As if is lv(x,t)l then O(x). earlier, Ila(.,t)lll sufficiently small, 0 < cl value :5 O(x) + a(x, t) :5 C2 < oo for this t and respectively there exists function a real-valued continuous in finite inan absolutely w(x,t), arbitrary terval and unique the term 27rm, m to it, such that up to adding 1, 2,..., Since v(-, t) E X' and vx'(x, t) v(x, t) (O(x) +a(x, t))e'1'_0t+'0(x,t)1. [0'(x) +a'(x, t) + we have i(O(x) + a(x,t))w.,(x,t)]e'l'-t+w(x,t)), E L2 if Ila(.,t)lll is sufficiently small. if u(x, t) : 0 for some t > 0 and all x E R, where By analogy, u(x, t) is a Let
a
-
=
=
-
=
-
=
=
=
X
X'-solution
of the
Theorem
of
tion
the
111.3.7
complex
corresponding for
if
X1, luo(-)l
uo G
any
ing X1 -solution onto
the
let
the
of
the
0(-) E H1, Ila(., O)l 11 u(x, t) of the problem (Iff. 0)
t >
the
following
0 such
that
ILO,1(*) 0) 12 < 6, then the correspondis global 3), (111.1.4) (it can be continued t > 0 one has I U(-, t) fixed 0(-) E H1,
6 and
(R (T))
1
'r='ro
of minimum
of the
function
"0
2
I
00
a(x,t)o'(x)[q
opera-
C1 Ig 1 22 for
proved.0
Therefore,
2
=
-w"
202 fl(02
+
spectral
the
(111.3.6),
t) 0
Oa)2f/((o
Mi(g)
that
Ao is the smallest
sign, from
=
=
operator
L with
operator
it follows
number
-r,
the
of
H' satisfying
The -
0
f(02 )
+
spectrum
is of constant
Hence,
tor
>
LW
operator
_Zj
=
of the
0'(x)
since
the
Consider
continuous
+
condition
the
(91 01) Proof.
2(0
-
(0, 1).
E
Lemma 111-3.9 g E
202fl(02)
ay +
f(02)
+
202f/(02
)]dx.
R('r)
STABILITY OF SOLUTIONS
CHAPTER3.
100
We take
jZj
sup
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
g E
H' sat,sfying
C2Ig 12,2
MI(g)
Lemma 111.3.10 the
C2
where
Cj(1
=
K) -2, for
+
all
real-valued
condition
W
I
g(x)o'(x)(q
Zj
-
+
(02(x))
f
+
202(X)f1(02(X)))dX
(111.3.7)
0.
=
00
Proof. g
=
ao'+
Represent
p)
(0',
o where
arbitrary
an
0.
=
function
Then M,
(g)
CIJW12.2
M, (W)
=
(111.3.7)
H1 satisfying
9 E
in the
form
from condition
We get
(111.3.7): 00
a
j
0/2rq
_
f(02)
Zj +
202f/(02
+
+j
)]dx
00
W01[q
_
ZU +
f(02)
+
202f/(02
)Idx
=
0,
00
-00
hence, co
f 001[-q I 1101 1 2 a
=--
101 12
f(02)
+
+
202f1(02)]dX
0
n
We define
IV.1.6.
measures)
Gaussian
dimensional
kind
Definition
from
measure
arbitrary
an
H of the
space
.Clearly,
0
constant.
centered
of Borel
sequence
follows.
115
Borel
sets
setting
and,
n
n
wn(M)
(27r)-'T
=
][I
n
L
Ai
2
i=1
obviously
we
Mn.
get
procedure
this
Repeating
Gaussian
dimensional Now
we
sigma-algebra
Borel
Hn
that
subset
I el,
span
=
each
Min .
-
-)
A n Hn
sigma-algebra Borel
M1 is
sigma-algebra M, obviously
subsets
because
by
all
and
open
Now,
separable
closed
let
the
and
only
SO,
(n
=
open
1, 27 3,...)
IVn}n=1,2,3....
C
:
and closed
be
weakly
real-valued
arbitrary
Lemma IV.1.10
the
Then,
c
--+
>
as
Borel
with
get
measures
Borel =
easily
can
we
the
1,
the
to
is
a
M,
set
M. But
then,
contradiction
containing
in H.
in
complete
a
We recall
1, 2,3,....
=
that
that
a
measures n
Borel
verify
sigma-algebra
converging
I W(x)vn(dx)
measure
v
in
Mif
W(x)v(dx)
M
n
H,
of
minimal
vn(M)
=
one
a
M}
A E
some
Consider
coinciding
subsets
nonnegative
v(M)
that
Then,
whole
A E Msuch
exists
supposition.
is the
A n Hn is
show that
A n Hn for
the
onto
if
lim
an
of finite-
A E M(we recall
Hn),
n
there
in Mand not
is called
n-oo
for
=
the
be considered
Wn can
Msuch
space
sequence
Then,
sigma-algebra
Borel
sets.
v, vn
metric
that
the
definition
sigma-algebra
sequence
our
extended
wn(A to
A n Hn E M1. n
that
all
=
suffices
it
Mn by
n
contains
get
we
the
on
naturally
wn(A)
C H
Mn7 M1 :
in H contained
a
0,
>
be
opposite.
fC
=
A of H such
of all
.3ince
the
n
and M1 C n
dx,,,
...
defined
wn
n
can
this,
To prove
M'
Clearly,
Mn.
Wn
rule:
the
by
H
Suppose
of Hn if A E M.
integer
measure
1).
en
dxj
jWn}n=1,2,3,.-
measures
show that
e
measure
all
for
1
2
F
additive
countably
a
Y>'-1.?
-1
Let
of
sequence
M
bounded
the
functional
continuous
measure
w
from Definition
jWn} weakly
measures
0 in M.
be
IV.1.6
converges
to
the
countably
measure
additive. w
in
H
as
oo.
Proof.
First
0 there
exists
of a
all,
one
compact
can
set
prove
If,
as
in the
C H such
proof
that
of Theorem
w(If,)
> I
-
e
IV.1.8 and
that
1,Vn(K,)
for > 1
any -
c
CLIAPTER 4.
116
integer
all
for
functional
let
Further,
0.
>
n
W defined
take
us
j
lim
W(x)w(dx).
W(x)w,,(dx)
H
Take also
arbitrary
an
>
E
one
easily
can
that
verify
there
6
exists
=
8(e)
>
0
that
such
IWW W(Y) I any
ifn
Let
=
if,
H,,,
n
Ix
satisfying
and y E H
If,
E
x
n
f
p(x)w,(dx)
so
that
f
-
I W(x) 1,
M= sup
0
n
<eM,
Kn
H
where
IH
Obviously,
1, 2,3,....
=
y
-
(IV.1.5)
0, such
show the
of C > 0,
existence
I
n-oo
W(x)w(dx)
in view of the
(IV. 1.4),
i.
Y E H:
Y
Then,
K,
C
the
e.
=
Kn,
arbitrariness
of
statement
of the
E
Hn7
sufficiently
all
I all
the
by
w,,
Hilbert the
large
sufficiently
product
rule:
0
I w
on
0
real-valued
a
of H.
subsets
C2
some
arbitrary
an
open
bounded
set
Q C H.
n-oo
(K)
liM SUP [In
:5
p(K) for
an
arbitrary
closed
bounded
K C H.
set
n-oo
Proof us
prove,
Fix
an
satisfying
example, >
0 and
c
following
the
0,(u)
0
equation tor
Borel
arbitrary
an
Borel
the
Then,
[to
-
class
it
is
of
(IV.2.5)
+
T]).
C(I;X).
eigenfunctions
Pn be the orthogonal
projector
Xn & Xn.
Consider
also
Xn
=
trans-
small
sufficiently T,to
invariant
an
val-
Therefore, Further, of the in the
the
let opera-
space
following
120
CHAPTER4.
problem
approximating 1
2
Un
+ Un,xx + Pn[f(X,
t
2
Un
_U
t
U
n
Let
pn
in the
space
Then,
the
the
Pn
0
0
Pn
)
1 n
,
X-P n[f(X,
gi
eigenelements
of the
(U2)2)U2]
n
0,
t E
R,
(IV.2.6)
(Ul)2
+
(U2)2)Ul]
n
0,
t E
R,
(IV.2.7)
n
(eO, 0),
=
is
(0, eO),
=
Clearly,
that
the
Xn is
the
onto
-
-
orthonormal
S.
nU2(X). 0
n
has a unique local solution (IV.2.6)-(IV.2.8) (as it is well known, in a finite-dimensional and
n
projector
92
an
operator
n
U2(X,to)=p
orthogonal
f9n}n=1,2,3,...
system
+
PnUI(X), 0
=
be the
(Ul)2 n
X
(X, to)
Let also
X.
(IV.2.1)-(IV.2.4):
problem
the
INVARIANT MEASURES
92n+1
7
-
basis
for
linear
=
(en) 0), the
in
X
space
integer
n
with
Xn 92n+2
(U1(X't)'U2(X'
=
any two
space
equipped
subspace
positive
any
un(x, t)
(IV.2.8) Xn (D Xn
=
(0, en)
n, the
n
norms
7
consisting
....
of
problem
t))
E C (I;
are
equivalent,
Xn)
X). In for 0 these solutions. 112 t) X for any n and for any uo Therefore, X the problem 1, 2, 3, (Ul0 (.), U20 has a unique global solution (IV.2.6)-(IV.2.8) Un(*) t) E C(R; Xn). it is clear the above solutions that Further, Un(*,t) of the problem (IV.2.6)the equations satisfy (IV.2.8) 1, 2, 3, (n we
mean
addition,
direct
the
space
shows
verification =
the
-dt-JlUn(*,
that
of the
norm
space
=
...
=
Un(', t)
=
A(t
_
to)pnUo
f
+
t
B(t
-
S)pn V(.' JUn(*, S)12 )Un(*) s)]ds.
(IV.2.9)
to
Hence,
u(., t)
(IV.2.5)
from
(IV.2.9)
and
one
has for
those
of t for
values
which
the
solution
exists:
JU(', t)
Un
-
(', t) I IX
:5
C1JJUo
pnUoJJX
_
+
I
C2
t
I JUn(*, S)
-
u(., s) I lxds+
to t
J
+C3
I JU(., 3)
pnU(.,
_
s) I Jxds.
(IV.2:10)
to
the
Here the
constants
solution
right-hand respect
u(-, t) side
to t E
C1, C2, C3 do
of this
[to,
for
exist
to +
t E
[to,
to +
T]
obviously
inequality
TI,
depend
not
therefore
we
on
where tends
get from
the
initial
value
T > 0.
Then,
to
as
zero
n
the -4
(IV.2.10)
inequality
to and t.
uo,
third
+00
term
in the
uniformly
with
by the Gronwell's
lemma that
lim n-oo
By analogy,
if the
max
tE[to,to+Tl
u(-, t)
solution lim n-oo
exists
max
tE[to-T,to]
JJU(',t)-Un(',t)JJX=Oon
a
segment
JJU(',t)-Un(',t)JJX=O-
[to
-
T, to],
Let
T >
0, then
ANINVARIANT MEASURE FOR THE NLSE
4.2.
121
Hence, lim for
all
fact
I
segments
implies,
tEI
of
the
it is easy
then
for
for
verify
to
any fixed
T2]
+
(c)
that,
if
u(-, t).
solution
and, hence,
IV.2.2
This
global
the
X.
u(., t)
function
a
of the
existence
function
t E R this
(IV.2.11)
0
=
of Theorem
uo E
any
t) I Ix
u,,(.,
-
of the
statement
(IV.2-5)
equation
Further,
(IV-2.5),
T1, to
-
particular,
in
solvability
[to
=
I Ju(-, t)
max
n-oo
is
G
C(R; X) of the
solution
a
satisfies
equation
following
equation:
T
u(.,,r)
A(T
=
t)u(-,
-
+IB(,r
t)
_
S) [f (.' I U(.' 5) 12) U(.' s)]ds,
R,
E
r
t
which, any
the
as
fixed
earlier,
for
the
map uo
t
transformation
(t
uo
fixed
any
has
t
u(., t) u(., t) as
is
--+
--+
Therefore,
X-solution.
from
map from
a
global
unique
a
one-to-one
X.
X into
X follows
X into
for
The
continuity
from
the
of
estimate
to)
>
t
U
('i t)
Vt)
-
X
C1
!
U
('; to)
V
-
(*) to)
X
+
C2
)rI
lu(.,,.s)
v(-, s) I Jxds,
-
to
u(., t)
where estimate
v(., t)
and
for t
0
c
all
numbers
n
=
T,to+T]
1, 2, 3,
(IV.2.6)-(IV.2.8),
problem
for
and
...
taken
I JUn(*, to)
(here
u,,(.,
to)
Proof
=
pnuo
follows
v,,(.,
and
from the
to)
=
estimate
-
same
(c)
of Theorem
8
exists
value
>
and
IV.2.2
0 such
a
similar
proved.
are
that
I JUn(i t)
max
tE[to
for
solutions
(a),(b)
the statements
u,,
(., t)
satisfying
n,
t) of
and Vn (*) the
the
condition
to) t
I jUn(',
t)
-
V.(-, t) I IX
!5 C1
I JU.(',
to)
-
Vn(*i to) I IX
+
C2II
JUn(', 8)
-
Vn(-, s) I Jxds,
to
that
results
from
By hn(UO) t)
Un(')
t +
to)
where
equation we
(IV.2.9),
denote
Un('i t)
the
is the
and
function solution
an
analogous mapping
of the
for
estimate
any
problem
uo
E
t
0
phase
the
with
system
space
Since
result
the
to
Borel
in the
S.
in X.
following
dynamical
a
consider
us
operator
in Xn
the
hn is
function
1, 2, 3,
=
INVARIANT MEASURES
-VaEn(a,
=
bi(to)
(t)),
b(t)
b),
(IV.2.13)
(U2 (-,to),ej)L2(0,A)i
=
(i==1,2,...,n),
n
(bo (t),
=
(IV.2.12)
bn (t)),
...'
un
(IV.2.14) b)
(Ul,
=
U2n ) and En (a,
n
A
f I! [(UI,X)2
E(Un)
2
(U2,J2]
+
n
F(x, (Ul)2
-
n
+
n
(U2)2))
Then,
dx.
n
according
The-
to
0
the
IV.1.3,
orem
dynamical
(W.2.12)-(W.2.14)
system
with
system possesses
Borel
a
phase
the
invariant
n
y' (A)
ii
-(n+l)
(27r)
=
n
A,
'=O
A C R2(n+l)
where
IV.1.1,
there
space
R2(n+l)
is
and
A C R2(n+l)
is
an
natural
a
Borel
arbitrary
Borel
Q of the
subsets to
Borel
a
e
set
ji':
measure
En(a,b)
n
db,
da
Further,
set.
space
according
between Xn defined
0 C Xn if
an
by
A of the
subsets
the
element
Proposition
to
Borel
rule:
to
9 when and
only
when
(a, b)
E A where
ul
Un
(Ul,
=
(ao,
...,a
correspond measure
n);
,
to
it,.
b
each
These
=
(bo,...,
other
bn).
In
in this
arguments
addition,
sense,
easily
then
aiej,
imply
if two sets
jUn(Q) the
=
A C
y' (A) by
statement
n
set
U2 ) ben
n
n
U2
E biei
and
i=O
'=0
a
Borel
a
n
longs
by the
generated
A
correspondence
one-to-one
corresponds
f
R2(n+l)
space
R2(n+l) the
and
definition
of Lemma IV.2.4.0
1
C X"
of the
ANINVARIANT MEASURE FOR THE NLSE
4.2.
According the
to
verges
measure
w as
n
Lemma IV.2.5
p(Q)
Proof.
arbitrary
the
IV.1.10,
Lemma
to
of Borel
sequence
measures
weakly
Wn
con-
oo.
--*
t)) for
tt(h(fl,
=
123
any
bounded
open
Q C X and
set
for
any
t E R.
Fix
according
Then,
set.
is open
4) is bounded
4.1, the
proved
too,
and
Q,
c
=
For any A C
Lemma
ly
IV.2.3,
for
B6(x)
x
and for
n
K,
of the
x).
y
-
B6,(xi)
be
the
since
Further,
< e.
h(K, t)
=
h(Q, t)
functional
from
of results
obviously,
Section
according
is
>
a
that
for
to
compact
a
set,
11hn(u,t)
covering
of the
According
0.
any
has
one
finite
a
Then,
aQ,)}
Then,
1,2,3,...
=
IV.2.1,
A and let
set
6 > 0 such
exists
any
B6, (xi),...,
Let
IV.2.2,
a.Q); dist(Ki,
yEB
E K there
x
61
bounded
open
of Theorem
X and in view
space
of Theorem
aA be the
X, let
dist(A,
again
of the
e
arbitrary
an
(a),(b),(c)
arbitrary
an
K C Q such
(a) and (b) h(f2, t).
a
(where
fix
us
set
Q C X be
statements
subsets
compact
a
statements
K,
Let
bounded
on
exists
proved
to(...
let
E R and
t
the
to
bounded,
and
there
an
u,
E
v
to
B6(x)
-h,,,(v,t)llx compact
0'
no.
n
y
of
:5 p (B) +
inf yn
< lim
n-oo
(because Hence,
p(Q)
yn(B) due
>
=
to
p(Qj).
open
an
Remark be
unbounded,
the
lim
=
R
arbitrary
proximation
p(Q)
Xn) =
proved
(B)
+
E
=
yn(hn(B of
tt(Qj),
c
n
Xn' t))
0,
>
(hn (B, t))
(c)
u
E B and
we
+
c
(f2j)
< y
+
c
hn(B n Xn' t) C hn (B, t)). By analogy ft(Q) < jz(,Qj).
and
have
and Lemma IV.2.5
statement
2
all
and IV.2.4
lim inf yn
2
n
>
2, A > 0, T
Hpner(A) to),
u(., t)
where
Eo,
...'
E,,,-,
>
Hpn ,JA)
into
by Theorem L1.5,
functionals
dynamical Let
the
IV.2.2
of
view
in
As
0.
>
of such
any
KdVE
u(x
this
on
of the
is dense
points
ut+uux+uxxx=O,
the
r
sense
f (x, s)
and
1+s
series
An infinite
tion
the
of Theorem "
X and
c
phase
new
a
By analogy,
R > 0.
of the
set
for
bounded
4D is
any
any
set
balls
case.
4.3
the
0
there
0
R
exists
and
d,
2 be
n
of equation
Lmk -solution
the
+oo
--
the
is
obviously
is
by
-dt-Ej(um)
=
u'(x,
solution
generated
(IV.3.4),(IV-3.5)
problem
the
is
1Um(*,tO)jL2(0,A)
=
of the
local
in L,,,
-dt-Eo(u')
that
verified
onto
(IV-3-5)
classical
unique
a
finite-dimensional
a
solution
Proposition be such
solution
be continued
can
addition,
the
(x).
uo
(the topology
1UM(*7t)1L2(0,A) for
P"'
=
INVARIANT MEASURES
the
(n
law
conservation
>
2):
A
2
Eo(u)
I(DnU)2 2
+ En(U)
X
1
+
2
u
2+ cnu(D'-'
X
U)2
-
Dn-2U)
q,,,(u,...'
X
dx >
0
1
211UG)11n where
n
mates
for
Ej(u)
(s)
is
have
1U1L2(0,A))1'_1
2
p
function
!Eo(u) 2
,
in view where
of the
-
and
continuous
functionals
the
we
a
2
+
?7n(JjU(-)jjn-1)) increasing
!Eo(u) 2
E2(u),...'
known
on
inequality
+
[0, +00). En-1 (u).
lUlLp(o,A)
Repeat For
1+1
'EO(U)
2
2
IJUI12 1
-
?71(1U1L2(0,A))(jJUJj12*
+
I)-
esti-
functional
JU12L2 (0,A) (jDxujL2(0,A) P
p > 2:
El (u) +
the
these
+
ANINFINITE
4.3.
We get
SERIES OF INVARIANT MEASURESFOR THE KDVE 127
by step
step
from
the
obtained
I I u 111 for
all
There
Lemma IV.3.5
(R, s)
on
[0, +oo)
E
satisfying
0
(d),...,
:5 C,
and Lemma IV.3.4
R,
t E
estimates:
proved.
is
n
functions
exist
[0, +oo),
C,,, (d)
u
(R, s),
-y,,
such
that
(R, 0)
-y,,,
nondecreasing
monotonically
in
=-
and
8
defined
0,
continuous
following:
the
d
Tt En(umk(*7t)),
X
0
By analogy
t)Umk (.' t))
0
X
proved.El of Theorem
IUMk (.' t)
-
1.1.5,
one
U(., t) I In
__
can
0
prove
that
for
any
t
0
CHAPTER4.
130 k
as
Hpn, ,,(A),
if uo C-
oo
--+
Corollary
IV.3.9
[E" (um (-, t
lim
Let
M-00
Proof IV.3.3.
the
repeats
be integer.
> 3
to
such
> 0
Theorem
prove
Proposition
Then,
Lemma IV.3.8
IV.3.10
IV.3.2,
For
any
>
n
for
Hpn,,-,'(A)
any uo E
and
t
any
in
view
of the
need
three
Proposition
proved
any
we
shall
also
2,
uo E
Hp'e,(A),
>
e
statements.
0 and t E R there
exists
that
I 1Um(-' 0 for
proved.0
is
0.
==
of
proof
IV.3.3
D
Below,
8
n
Proposition
(um (., to]
E"
-
and
INVARIANT MEASURES
m
and
1,2,3,...
=
UN., 0 11.
-
arbitrary
an
0
be
En (Um (*
7
t)
and
__+
[0, +oo)
:5 77n
Hpn,-,'(A).
Lemmas IV.3.5
with
Then, on
to)) 1
0
==
ul). k
as
uo
and, u
--+
Then,
there
such
oo,
IV.3.6.13
=
also,
uT"
and
that
that
for
IV.3.10
any
such
(I I Um(*, to) I I
t
1)
+oo
--+
--+
U(.,t)
proved.0
E R there
that
n-
is
Mk-
m=
mA;
(-, t) andUMk(.,t)
Proposition
integer.
i
uo
-+
> 6,
contradiction.
> 3
11,,
uo
is unbounded
UMk (. 1
IV.3.3 a
with
Pm,
-
(W.3.4),(W-3.5)
and continuous
and uo E from
UMk(.'
-
sequence
we
nondecreasing
follows
7
by Proposition i.
I I uml'
where
problem
of the
I En (Um (') t)) for
0 such
>
c
(W.3.4),(W.3.5)
problem
of the
and
solution
is the
exists
Jjumj(-,to)-um(-,to)jj, 0 and t E R there
+11PM'(UM(-'s)UXM(-'s))111) for
m --+
similar
in
second
the
IV.3.1,
zero
of the
I Pm' [D'
max
0:5"35--l i+3:pd2n-2
UoEB,(U)
to
side
Estimating
m.
0 and
sup
=
right-hand
(IV.3.4),(IV.3.5)
problems
the
+
U.
arbitrary
for
IV.3.10
in the
term
of
+
i
X
tend
inequality
of this
side
uo
=
-9) 1 L2 (0,A)
D1X-9(.,
x
S) I L2 (0,A)
D' Um(.,
x
Di U(-, s)] 1 L2 (0,A)
x
and Theorem
IV.3.3
by Proposition
Further,
uo.
with
Propositon
Dx%-, s)
-
solutions
the
are
(-, s)
D'i!m X
s)
X
and
In view
s)
mo
m
Fix
these
place
takes
if
uo E K if
Hpn,-,'(A)
wi
of
(I
+
Bjuj).
A '-'),
+
point
B, (ul)
-,
where
i
finite
covering
a
any i the
for
(IV.3.6)
jek}
K
ball
a
of the
K
set
(IV.3.6)
relation
is also
the
is
S of the
change
of
valid
for
all
basis
in
above-indicated
kind
(i.
I in the
definition).
by
n
orthonormal
n
-
Then,
0, 1, 2,....
=
set
proved.0
is
operator the
compact
obviously
From here
with
of the
be
that
such
IV.3.12
of the
functions
A' )-'(l
-
.
Then,
3.
>
n
eigenvectors
above-defined =
B, (ul),
Proposition
integer
consisting
also
Let
mi
and
each
to
be numbers
and uo E
arbitrary
the
are
el,
ml,...,
> rni i
an
r.
Let
property.
> maxmi,
m
Fix
e.
m
f-
and let
balls
V1 1,,_1
-
the above
possessing by
I luo
if
arbitrary
an
INVARIANT MEASURES
eigenvalues
are
wi
of
S. Consider the
subspaces
the
in
dimensional
finite-
Lk
Gaussian
Hpn,-'(A)
C
measures
Wk
Wk(Q)
(27r)
=
2A;+1II
Wi
Q
Then,
W1, is
the
a
measures
sian
I [(Ui eO)n-1)
Lk
Borel Wk
the
IV.1.10
E
u
7
be considered
fWk}k=1,2....
Wn For
Borel
a
7
.
any
the
functional
of this
subsets
J,,
weakly
the
space,
(IV.3.4),
(IV. 3.5)
and any
uo E
taken
tl,
at
Hpn,-,'(A)
L
so
of time
moment
e
I,
into
according
Lemma IV.3.13
lim Proof.
(ZO (t)
7
Let
us
(t))
where
Z2m
and the
h,,, (.,
rewrite u-
(x, t)
also
set
HPn,;1(A),
sets.
Section
by
and
4.1,
Lemma
infinite-dimensional
phase
t) (-,
Gaus-
rule
any
t +
to)
h,,, (u, t)
-
t).
i(t)
a
=
(IV.3.4),(IV.3.5) + Z2m (t) zo (t) eo (x) + =
JV-,H(z(t)),
-
-
u,
fixed
bounded Let
the
system
L..
into
L..
(IV. 3.4), (IV.3.5) t transforms
bounded
closed
(Q))
it,,,
-
problem
h,,, (P.,,
system =
of the
a
be
by
t transforms
h,,, (., t) for =
W2'-'
fixed
on
HPn,,,1(A).
in
generated
L,,,
space
for
and bounded
well-defined
are
_
Obviously
(p,,, (h, (Q, t))
the
/,n
to.
Let t E R and Q C
M-00
the
Borel
are
from
results
to
in
Hpne-,' (A)
in
u-
the
to
we
solution
t +
&2ki
...
F C R2k+1
and
to
converges
ltk
with
function the
maps into
Fj
measures
continuous
system the
that
Idzo
e_j'(u)dWk(U)-
measures
h, (u, t) be the dynamical
jei}i=o,1,2'...'2k
rule
Z? E W` i
According
k.
Hpn,,(A)
Q C
obviously
is
the
_0
e
E
Borel
Ak (f2)
Since
2
(U, e2k)n-1] as
set
by
F
in Lk for
measure
can
sequence
measure
...
1
f
I -
i=O
where
defined
vectors
over
2k
2k -
spanned
set.
also
Then
0.
in
the
e2m
(X).
coordinates
Then,
we
z(t) get
(IV.3.7)
SERIES OF INVARIANT MEASURESFOR THE KDVE 133
ANINFINITE
4.3.
(to)
z
H(z) matrix (i. 1,2,...,m)and where
Let
::::::
+
J*
0 for
det(
that
prove
Lebesgue
the
IV.1.3,
=
J is
(1 (jrk)2n-2) +
A
)
azo,3
f
orm(Q)
k,1
indexes 1 for
ij=5_,_2m
measure
(2m + 1) x (2m + 1) (k -(J)2k,2k-I
=:
A
of the
values
2aut
(IV.3.8)
2m,
0,
skew-symmetric
a
_2irk
=
other
all
i
and
+
---
(J)k,l
us
Theorem
-2 2me2m) -A (J)2k-1,2k
Ej(zoeo
e.
(uo, ei),,-,,
=
dzo
all
0,1,...,2m.
according
Indeed,
t.
dZ2,,,
...
=
is
invariant
an
to mea-
n
for
sure
dynamical
the
(IV.3.7),(IV.3.8).
(hm (Q, t))
I
=
h-
arbitrary
an
this
Borel
immediately Let
generated
L,,
space
by
the
problem
Therefore,
orm
for
phase
the
with
system
arguments,
dZ2,rn
...
dzo
...
dZ2m
(O,t)
V =-:
that
of the
In view
of the
continuity
17,
function
1.
closed
arbitrary
an
Vdzo
dZ2m
...
9 C R2m+1
set
implies
take
us
dzo
bounded
Hp'e-r'(A).
QC
set
In view of the
above
get:
we
ym (hn
(Pmu) -E,, (hm (u,t))
eEn
(9, t))
dyn (u).
Further, ym (Q)
therefore,
according
integrand
in the to
KC
Hpne-r'(A)
proof
m
of Theorem
side
equality
E Q.
u
p(Q \ K)
c
of which
existence
By Proposition
IV.1.8.
and
IV.3.11 of this
0 and
>
that
such
jn(P u)-En(hm(u,t))
(Q, t))
Proposition
to
right-hand
integer
respect
pm (hm
-
0 and
can
be
obtain
the
that
uniformly a
with
compact
proved
as
set
in the
IV.3.12, -
tt,,
(h,,, (K
n
Q, t))]
=
0,
M-00
hence,
by Proposition
IV.3.11,
we
lim sup
get the relation
[ftm (Q)
-
ym (hm
(Q, t))]
bounded
I tt,, (Q)
-
0, yields
open
p,,,
(h
the
set
statement
0 c
(Q, t)) I
Hpn,-'(A) =
0.
of Lemma IV.3.13.0
and
for
any
t E R
134
CHAPTER4. Lemma IV.3.15
,,n(Q)
,n
=
By Theorem
IV.3.1
set,
B)
Clearly, E
v
that
,n
inf
=
and
(0 \ K) < c. f2j. (Q, t) JJU Vjjn-j
these
u
-
balls.
any
also
Let
P
=
aA is
and
B1,
E
v
of
Proposition
r'(A),
for
sufficiently
all
,,n(f2)
,n
Corollary
(B, t))
+
IV.3.14 e
c
uEA, vEB > 0.
a
for all
bounded
a
Lemma IV.3.4
arbitrary
an
and K, C h n-1
too,
dist(A,
Take
too.
K C Q such
be
(Q, t)).
n-1
Hpn,,-,'(A),
in
set
Q C
(h
Proof.
Hpn,,-,'(A)
Let
INVARIANT MEASURES
=
,n(Qj)'
n
IV.3.2.
First,
let
Q C
Hpn,,-,'(A)
be
an
open
(generally
We set
f2k
=fuEQ:
11h
n-1
(U, t) I 1 _j
+
jjujj.,,-j
0.
Then
U Qk
Q
,
and
each
set
f2k
is
open
and
k=1
bounded;
in
addi-
00
tion,
hn-1
(n, t)
U
h n-1
(Qk, t)
and
/,,n(gk)
.
,n
(h
n-1
k=1
(Qk' t)) by
Lemma IV.3.15.
Therefore,
,,n (h
n-1
(n, t))
=
liM k-00
Un (h
n-1
(Qk' t))
=
liM k-oo
ttn
(f2k)
=
,n
(f2).
135
ADDITIONAL REMARKS
4.4.
Let
hn-1
(A, t)
now
can
last
is
a
Eo,...,E,,_1
all,
of
trajectories based
not
[16,20,53,66]).
Here
Concerning Some of them
[105]
in
an
with
the
sociated
invariance In
Poincar6
consider
there
equation.
is
nonlinear are
papers
for
with
quite
different.
In
the
and A
for
finite
for
any
f(X, S) :5 C(l p E (0, 2) for N
=
result
this
the 1 in
our
ball
The obtained
B C X.
+
S
d2)
A > 0.
for
is obtained
problem
all
for
by an
x,
>
J.
=
paper
initial-boundary with
Bourgain
arbitrary
>
A.
initial
[16,17] This
seem
to
data
this
allowed
AJuJP
for
in these
exploited
two
explicitly
more
this
a
author
AluIPu
and
to
space
be
p >
question
sufficient
and
nonzero
I_L(B) < -C(l+s di.)
00
:
0 if A < 0 and remains
(IV.2.1)-(IV.2.4)
open
with
L2_ The required
well-posedness of this
0
p >
0
