PII. S0161171201011450 http://ijmms.hindawi.com © Hindawi Publishing Corp.
ON AN INFINITE SEQUENCE OF INVARIANT MEASURES
FOR THE CUBIC NONLINEAR SCHRÖDINGER EQUATION
PETER E. ZHIDKOV
(Received 21 January 2001 and in revised form 28 May 2001)
Abstract.We consider the Cauchy problem periodic in the spatial variable for the usual cubic nonlinear Schrödinger equation and construct an infinite sequence of invariant mea-sures associated with higher conservation laws for dynamical systems generated by this problem on appropriate phase spaces. In addition, we obtain sufficient conditions for the boundedness of the measures constructed.
2000 Mathematics Subject Classification. 35Q55, 37K99, 46N20.
1. Introduction. Consider the Cauchy problem periodic in the spatial variable for the cubic nonlinear Schrödinger equation
iψt= −ψxx+2κ|ψ|2ψ, x, t∈R, (1.1)
ψ(x+A, t)=ψ(x, t), (1.2)
ψx, t0
=ψ0(x). (1.3)
Hereiis the imaginary unit,t0∈R,ψ=ψ(x, t)is an unknown complex function,ψ0 is a complex function periodic inxwith the periodA >0 andκ=1 orκ= −1. As it is well known, (1.1) supplied with condition (1.2) is formally a completely integrable Hamiltonian system possessing an infinite series of conservation lawsQn(ψ)which
are real functionals quadratic with respect to the highest derivatives of the function ψformally satisfying the propertyd/dtQn(ψ(·, t))=0,n=0,1,2, . . .(seeSection 2).
Further, it is known that in the finite-dimensional case to any conservation lawQ(p, q) of a Hamiltonian system of the kind
dp(t)
dt = ∇qH(p, q),
dq(t)
dt = −∇pH(p, q), (1.4)
whereH(p, q)is a smooth real function,nis a natural number, andp(·), q(·)∈Rn,
there corresponds a family of invariant measures (IMs) with densities f (Q(p, q)) wheref (·) is an arbitrary smooth real function. The problem we are interested in is whether this property is kept for the infinite-dimensional problem (1.1), (1.2), and (1.3), that is, whether conservation lawsQn(ψ),n≥2, generate IMs, too.
it. However, the invariance easily follows from the results presented in this paper. An IM for a physical system is considered in [6]. In [11], an IM is presented for a problem periodic in the spatial variable for the nonlinear wave equation
utt−uxx+u3=0. (1.5)
Unfortunately, some details of the proof seem to be not completely satisfactory in this paper. In [24], a result on IMs is presented for an NLS with a weak nonlinearity. Paper [26] contains an extension of this result to a nonlinear wave equation. In [27], a construction of an IM is presented for an abstract DS such that a lot of “soliton” equations are reducible to that form. Simultaneously, a result for a nonlinear wave equation similar to the above-described was proved in [18]. In [2], the results from [27] are reconstructed; the author established more careful proofs of results from [27].
All the above-indicated papers dealt with an NLS or a nonlinear wave equation (ex-cept [11]) contain results on IMs under severe constraints for nonlinearities. So, the problem appeared how to extend this approach onto a wider class of nonlinearities. Treatments of this problem are made in [5,16,25,28,29]. In [5], Bourgain, using his result from [3] on the existence ofL2-solutions for the Cauchy problem periodic in the spatial variable for a one-dimensional NLS with the power nonlinearityκ|ψ|p−1ψ, constructed a bounded IM forκ= −1 and 1< p <5. Similar results are contained in [16,25,28] (with 1< p <3 ifκ= −1 and 1< p <∞ifκ=1 in [25]) but in these papers suitable results on the well-posedness of initial-boundary problems for equa-tions under consideration are not proved and are contained as hypotheses. In further publications Bourgain [5] constructed also IMs for a multidimensional NLS and, in [5], considered an IM for the Korteweg-de Vries equation.
Papers [1,2,5,6,11,16,18,24,25,26,27,28] (except [18]) concern only with an IM associated with the concrete conservation law, the energy. At the same time, now it is known that certain evolution equations such as the Korteweg-de Vries equation or the cubic NLS possess countable sets of conservation laws and are formally infinite-dimensional Hamiltonian systems. Therefore, since in the finite-infinite-dimensional case any conservation law of a Hamiltonian system leads to a family of corresponding IMs, the question naturally arises whether to higher conservation laws there correspond IMs in some infinite-dimensional cases. In [9], this question is considered for a discrete system of Moser-Calogero particles. A construction of an IM corresponding to a higher conservation law for the sinh-Gordon equation is contained in [18]. In [29], it is proved for the Cauchy problem periodic in the spatial variable for the Korteweg-de Vries equation that to any known conservation law of the kind
Ln(u)=
A
0 1
2
u(n)x
2 +qn
u, . . . , u(nx−1)
dx, (1.6)
wheren≥3 is an integer, there corresponds an IM for a DS generated by this equation. In the present paper, we continue the investigations began in [29] and present an infi-nite series of IMs for the problem (1.1), (1.2), and (1.3). This result is also reestablished (without proofs) in [30].
nonlinear partial differential equations (in [7, 8, 15], the problem of the invariance is not considered). Second, at the time when “soliton” equations began to be inten-sively studied, there arose a question known as the Fermi-Pasta-Ulam phenomenon (see [20]) which, roughly speaking, consists in the stability according to Poisson of all trajectories of a DS generated by a “soliton” equation. If one has a bounded IM for that DS (so that the measure of the whole phase space is finite), then the Poincaré recur-rence theorem gives a (partial) explanation of this phenomenon. In this connection, it is important to note that, due to the problem (1.1), (1.2), and (1.3) being completely integrable, the method of the inverse scattering problem may give a more complete information about the recurrence and other properties of solutions. For example, such investigations have been performed by McKean and Trubowitz [17] forC∞-solutions and Bourgain [4] forL2-solutions periodic in the spatial variable of the Korteweg-de Vries equation. In these papers, the almost periodicity of solutions is proved. At the same time, the author of the present paper does not know rigorous mathematical results implying theXn-recurrence of solutions of the problem (1.1), (1.2), and (1.3)
given byTheorem 2.5.
2. Notation, preliminaries, and main results. In what follows, by C, C1, C2, C, C, . . . we denote positive constants. Everywheret and x are real variables. We fix a positive integernandA >0. LetL2be the usual Lebesgue space consisting of com-plex functionsu(x)of the argumentx periodic with the periodA, equipped by the scalar product
(f , g)= A
0 f (x)g(x) dx (2.1)
and the normf =(f , f )1/2. LetC∞be the set of complex functionsu(x)periodic with the periodAand infinitely differentiable. Let∆be the closure in the spaceL2of the operator−d2/dx2defined first on the setC∞. It is well known that∆is a nonneg-ative selfadjoint operator in the spaceL2. Further, letnbe a nonnegative integer and letHnbe the usual Sobolev space which is the completion of the spaceC∞taken with the scalar product
(u, v)n=(u, v)+
dnu
dxn, dnv
dxn (2.2)
and the normun=(u, u)1n/2. In fact,L2=H0andHn are Hilbert spaces forn= 0,1,2, . . . .LetDn
x=dn/dxnorDnx=∂n/∂xnandDx=D1x. Finally, consider a Banach
spaceXwith a norm · X and letIbe a connected subset of the real lineR. Then,
byC(I;X)we denote the Banach space consisting of bounded continuous functions fromIintoX, with the normu(t)C(I;X)=supt∈Iu(t)X.
the cylindrical set if and only if
M=f∈H|f , e1
H, . . . ,
f , ek
H
∈F (2.3)
for some integer positivekand a Borel setF⊂Rk. For a cylindrical set of the above
kind let
w(M)=(2π )−k/2
k
i=1 ω−1i /2
F
e−(1/2)ki=1ω−i1z2i dz
1···dzk. (2.4)
Then, it can be easily verified that the set of all cylindrical sets is an algebra and that the functionwis an additive measure on this algebra. In addition,w(H)=1. Further, it is known (see [10] for proofs) that the assumption that the operatorS is of trace class provides the countable additivity of the measurewon the above algebra of all cylindrical sets. Therefore, this measure can be uniquely extended onto the minimal sigma-algebraᏹcontaining this algebra. In fact,ᏹis the Borel sigma-algebra in the space H (for proofs, see [10]). The measure w considered on the sigma-algebraᏹ is called a centered Gaussian measure in the spaceH. It is essential for us that any centered Gaussian measure is a Radon measure, that is, for any Borel setΩ⊂Hand for any >0 there exists a compact set K⊂Ωsuch thatw(Ω\K) < . Finally, the following result is also useful for us (for the proof, see [27]).
Statement1. For any ballBr(a)= {u∈H| u−aH< r}, wherea∈Handr >0,
one hasw(Br(a)) >0.
The problem (1.1), (1.2), and (1.3) can be rewritten in the equivalent real form for real functionsuandv, whereψ=u+iv, as follows:
ut= −vxx+2κ
u2+v2v, x, t∈R, vt=uxx−2κ
u2+v2u, x, t∈R, u(x+A, t)=u(x, t), v(x+A, t)=v(x, t),
ux, t0
=u0(x)=Reψ0(x), v
x, t0
=v0(x)=Imψ0(x).
(2.5)
We suppose thatψ=ψ(x, t)is a solution of the problem (1.1), (1.2), and (1.3) infin-itely differentiable with respect toxandtand(u(x, t), v(x, t))is the corresponding solution of the problem (2.5). Letw1=ψ(x, t)and
wn+1= −idwdxn+κψ¯
n−1
k=1
wkwn−k. (2.6)
We set
Ln(ψ)=Ln(u, v)=
A
0 ¯
ψ(x, t)wn(x, t) dx, (2.7)
conservation laws for the problems (1.1), (1.2), (1.3), and (2.5)). Let
Qn(ψ)=Qn(u, v)=
1
2ReL2n+1(ψ), (2.8) wheren=0,1,2, . . . .Then, one can easily get from (2.6) and (2.7) that
Qn(ψ)=Qn(u, v)=
A
0 1
2D
n xψ
2 +1
2|ψ| 2−q
n
ψ,ψ, . . . , D¯ n−1
x ψ, Dnx−1ψ¯
dx, (2.9)
where the functionsqnare polynomials; in particular, they are infinitely differentiable.
In what follows, we writeQn(ψ)=Qn(u, v)and
qn
ψ,ψ, . . . , D¯ n−1
x ψ, Dnx−1ψ¯
=qn
u, v, . . . , Dn−1
x u, Dxn−1v
(2.10)
for simplicity of the notation.
Methods of investigation of the well-posedness of the problems (1.1), (1.2), (1.3), and (2.5) are proposed in a large number of articles (cf. [12,14,23]). In our one-dimensional case (x∈R1) these problems become essentially simpler. Now we formulate the result we need. Its proof is presented later.
Theorem 2.1. For any positive integer nand for anyψ0∈Hn andT >0there
exists a unique solution to the problem (1.1), (1.2), and (1.3) of the class ψ(·, t)∈ C([t0−T , t0+T ];Hn). For any fixedtthe functionψ0→ψ(·, t)is a homeomorphism as a map fromHn intoHn. Further, fn
1(f1n(ψ, t), τ)=f1n(ψ, t+τ) for allψ∈Hn andt, τ∈Rwheref1n(ψ0, t)=ψ(·, t+t0). In addition, the quantitiesQ0(ψ(·, t)), . . . , Qn(ψ(·, t))are independent oft(i.e., they are conservation laws).
Remark2.2. Sometimes we call solutions given byTheorem 2.1theHn-solutions
of the problem (1.1), (1.2), and (1.3) (or (3.1), see further).
LetHRen be the subspace of the spaceHnconsisting of real functions fromHnand letXn=HRen ⊗HRen be the Cartesian product of two samples of this new space. We equip the spaceXnby the usual scalar product
u1, v1
,u2, v2
n=
u1, u2
n+
v1, v2
n. (2.11)
Then, we introduce a selfadjoint operatorSnof trace class acting in the spaceXnby
the rule
Sn=
T−1 0 0 T−1
, (2.12)
whereT =(I+∆n+1)(I+∆n)−1. Take an arbitrary integern≥1 and letwn be the
centered Gaussian measure with the correlation operatorSnin the spaceXn. Now we
can give the following new formulation ofTheorem 2.1.
Theorem2.3. Let(u0, v0)∈Xnwheren≥1is an integer number. Then, for any
thenfn(fn(u, v, τ), t)=fn(u, v, t+τ)for all(u, v)∈X
nandt, τ∈R. In addition, the quantitiesQ0(u(·, t), v(·, t)), . . . , Qn(u(·, t), v(·, t))are independent oftfor these solutions (i.e., they are conservation laws).
In what follows, we set w(x, t) = (u(x, t), v(x, t)) for an arbitrary solution (u(x, t), v(x, t))of the problem (2.5). Not discussing various definitions of this con-cept, we accept thatfn(w, t)is the DS with the phase spaceX
n. We only remark that if
there exists a bounded IMνfor this DS so thatν(Ω)=ν(fn(Ω, t))for anyt∈Rand a
Borel setΩ⊂Xn, then the Poincaré recurrence theorem takes place for the introduced
DS (for details, see [19,27]). (It should be remarked that, since the functionfn(·, t)is
a homeomorphism as a map fromXnintoXn, it transforms any Borel subset of the
spaceXninto a Borel one.) For a Borel setΩ⊂Xnlet
µn(Ω)=
Ω
eΦn(u,v)dwn(u, v), (2.13)
whereΦn(u, v)=
A
0qn+1(u, v, . . . , Dnxu, Dxnv) dx.
Our main result is the following.
Theorem2.4. For any positive integern,µnis a Borel measure well defined in the
spaceXnand it is an IM for the DSfn. In addition,
0< µnDR
<∞ (2.14)
for all sufficiently large valuesR >0whereDR= {w∈Xn| |Q0(w)|< R, . . . ,|Qn(w)|< R}. Thus, the setsDRcan be taken for new phase spaces of the DSfn.
We remark that we do not study the question whether the measure of the whole phase space is finite, that is, if it is right that µn(X
n) < ∞, but we present in Theorem 2.4 a weaker statement sufficient for our aims. The following result is a corollary ofTheorem 2.4,Statement 1from this section and the Poincaré recurrence theorem (see [19,27]).
Theorem2.5. For any positive integernconsider the DSfn. Then, a.a. points (with
respect to the measurewnorµn) of the phase spaceX
nare stable according to Poisson. The set of points of the spaceXnstable according to Poisson is dense inXn.
3. Proof of Theorem2.1. We use standard methods of investigating the well-posed-ness of the Cauchy problem (1.1), (1.2), and (1.3) as in [12,14,23]. The idea of these methods is the replacement of the problem (1.1), (1.2), and (1.3) by the following abstract integral equation:
ψ(t)=e−i(t−t0)∆ψ0−2iκ
t
t0
e−i(t−s)∆ψ(s)2ψ(s)ds. (3.1)
Hereψ(t)is an abstract unknown function defined in segmentst∈I=[t0−a, t0+b], wherea, b >0, with values in a functional space (to be interpreted asHn). Formally,
Definition3.1. We call solutions of (3.1) (generalized) solutions of the problem
(1.1), (1.2), and (1.3).
We turn to provingTheorem 2.1. First, letψ0∈H1, then the existence and unique-ness of a global (defined for allt∈R)H1-solution to (3.1) can be proved as in [12,14] where the Cauchy problem (1.1), (1.2), and (1.3) is considered with initial data vanish-ing as|x| → ∞; in addition, it can be proved as in these papers thatQ0andQ1are conservation laws forH1-solutions. The proof of the existence and uniqueness of a localHn-solution to (3.1), wheren≥2 is integer, repeats the procedure from these
papers. So, we have that, for any integern≥2 andψ0∈Hn, there exista >0 and b >0 such that (3.1) has a unique solutionψ(t)∈C([t0−a, t0+b];Hn)and either this solution can be continued for allt > t0(resp., for allt < t0) or there existsb0>0 (resp., a0>0) such that this solution can be continued onto the half-open interval[t0, t0+b0) (resp., onto the half-open interval(t0−a0, t0]) and lim supt→t0+b0−0ψ(t)n= ∞(resp.,
lim supt→t0−a0+0ψ(t)n= ∞). In what follows, we show only that ourH
n-solution
ψ(t)can be continued onto the whole half-line[t0,+∞)because the statement that it can be continued onto the half-line(−∞, t0]can be proved by analogy.
Letψ0∈Hnand+∞ =T1≥T2≥T3≥ ··· ≥Tn>0 be such numbers that[t0, t0+Tk)
are maximal half-open intervals of the existence ofHk-solutions of (3.1). For the above
goal, it suffices to prove thatT1= ··· =Tn. The following estimate easily follows from
(3.1):
ψ(t)k≤C1ψ0k+C2
sup
s∈[t0,t]
ψ(s)l t
t0
ψ(s)kds, (3.2)
wherel=k−1 ifk≥2 or l=1 ifk=1,C1>0 is independent oft andC2(s)is a positive continuous function on the half-line 0≤s <+∞. The inequalities (3.2) easily imply the equalityT1= ··· =Tn. Thus, the existence and uniqueness of a globalHn
-solution (n≥2) are proved.
LetH= ∩n≥1Hn=C∞. Then, we have proved that for anyψ0∈H there exists a unique globalC∞-solution to (3.1). It can be shown as in [14] that theseC∞-solutions satisfy the problem (1.1), (1.2), and (1.3) and are infinitely differentiable with respect tot, too.
Lemma3.2. For any positive integernandd >0there existsR >0such thatgn<
Rfor allg∈Hnsatisfying the conditions
Q0(g)< d, . . . ,Qn(g)< d. (3.3)
Proof. We use the known embedding theorem
gLp≤Cg
1/2+1/pD
xg+g
1/2−1/p
(p≥2) (3.4)
which takes place for allg∈H1withC >0 independent ofg. Hence, using the func-tionalQ1(g)+Q0(g)=(1/2)
A
0{|Dxg|2+|g|2+κ|g|4}dx, we get the inequality
1 2g
2
1−Cg3g1≤Q
1(g)+Q0(g)≤ 1 2g
2
which implies that for anyd >0 there existsR >0 such thatg1< Rfor allg∈Hn
satisfying the condition|Q1(g)| + |Q0(g)|< d. Further, using (2.6), (2.7), and (2.9), we get
qn
ψ,ψ, . . . , D¯ n−1
x ψ, Dxn−1ψ¯
=q11
n
ψ,ψ, . . . , D¯ n−2
x ψ, Dxn−2ψ¯
Dn−1
x u
2
+qn12
ψ,ψ, . . . , D¯ nx−2ψ, Dxn−2ψ¯
Dxn−1uDxn−1v
+q22
n
ψ,ψ, . . . , D¯ n−2
x ψ, Dxn−2ψ¯
Dn−1
x v
2
+qn1
ψ,ψ, . . . , D¯ xn−2ψ, Dxn−2ψ¯
Dnx−1u
+q2
n
ψ,ψ, . . . , D¯ n−2
x ψ, Dxn−2ψ¯
Dn−1
x v
+qn0
ψ,ψ, . . . , D¯ xn−2ψ, Dxn−2ψ¯
.
(3.6)
Hence, the inequalities
Qk(g)≥
1 2g
2
k−γkgk−1 (3.7) take place fork=2, n(hereγk(s)are positive continuous functions). Now the
state-ment ofLemma 3.2follows from inequalities (3.5) and (3.7).
So, the functionalsQ0, . . . , Qnare conservation laws forH-solutions of the problem
(1.1), (1.2), and (1.3). Further, the inequality (n≥1)
ψ(t)−ϕ(t)n≤C1ψ0−ϕ0n+C2
max
s∈[t0,t]
ψ(s)l; max
s∈[t0,t]
ϕ(s)l
× t
t0
ψ(s)−ϕ(s)nds
(3.8)
(herel=n−1 ifn≥2 orl=1 ifn=1) following from (3.1) for arbitrary twoHn
-solutionsψ(t)andϕ(t)(whereϕ(t0)=ϕ0) of this equation, in view ofLemma 3.2 and the continuity of the functionalsQ0, . . . , QninHnimplies two corollaries:
(A) forn≥1 the functionalsQ0, . . . , Qnare conservation laws forHn-solutions of
(3.1);
(B) Hn-solutions of (3.1) continuously depend onψ0, that is, for anyψ
0∈Hn, >0 andT >0 there existsδ >0 such that ifψ0−ψ0n< δ, then maxt∈[t0−T ,t0+T ]
ψ(t)−ψ(t)n< whereψ(t)is the solution of (3.1) withψ0=ψ0. Finally, ifψ(t)is aHn-solution of (3.1) withψ
0=ψ(t0)(heren≥1), then for any fixedt∈Rthe functionψ(τ)satisfies
ψ(τ)=e−i(τ−t)∆ψ(t)−2iκ τ
t e
−i(τ−s)∆ψ(s)2
ψ(s)ds. (3.9)
Therefore, the function ψ(t0)→ψ(t) is a one-to-one transformation of the space Hnfor any fixed t, hence it is a homeomorphism because it is continuous with the
4. An approximation of the NLS. In this section, we consider Galerkin approx-imations of the problem (1.1), (1.2), and (1.3). Here, among more or less standard results, we obtain certain statements crucial for us. These are Lemmas 4.4and4.5 andProposition 4.7. In this section, all estimates with the use of the Gronwell lemma are made fort > t0. Estimates fort < t0can be made by analogy.
Let
e0(x)= 1 √
A, e2k−1(x)=
2
A1+λ2k−1nsin 2π kx
A ,
e2k=
2 A1+λ2k
ncos
2π kx A ,
(4.1)
whereλ0=0,λ2k−1=λ2k=(2π k/A)2(k=1,2,3, . . .)are the eigenvalues of the
opera-tor∆. Then,{ek}k=0,1,2,...is the orthonormal basis in the spaceHnconsisting of
eigen-functions of the operatorTwith corresponding eigenvaluesωk=(1+λkn+1)(1+λnk)−1 (k=0,1,2, . . .). LetPmbe the orthogonal projector in the spaceHnonto the subspace
span{e0, . . . , e2m}. Let alsoXm=PmHn.
Consider the following sequence of problems:
iψmt = −ψmxx+2κPmψm
2
ψm, x, t∈R,
ψm
0 =ψm
·, t0
=Pmψ0 (m=1,2,3, . . .),
(4.2)
or, equivalently, in terms of real functionsumandvmwhereψm=um+ivm:
um
t = −vxxm+2κPmum2+vm2vm,
vm
t =umxx−2κPm
um2+vm2um, (4.3)
um0 =um
·, t0
=Pmu0, v0m=vm
·, t0
=Pmv0. (4.4)
First of all, we recall that in a finite-dimensional linear space any two norms are equivalent. Further, for any positive integer mand ψ0∈Hn obviously there exist a, b >0 and a unique infinitely differentiable solutionψm(x, t)to the problem (4.2)
defined for(x, t)∈R×(−a, b)and eitherb >0 is finite (resp.,a >0 is finite) and lim supt→t0+b−0ψ
m(·, t)
n= +∞(resp.,a >0 is finite and lim supt→t0−a+0ψ m(·, t)
n
= +∞) or the solutionψmcan be continued onto the whole half-linet > t
0(resp., onto the half-linet < t0). Then, the direct verification shows that
d dtQ0
ψm(·, t)=0 (4.5)
for allt. This equality yields, in particular, that for any positive integermandψ0∈Hn there existsC >0 such that
ψm(·, t)
n≤C (4.6)
Lemma4.1. For an arbitrary segmentI=[t0−T , t0+T ], whereT >0, and for any
positive integernthere exists a functionβn(s)continuous and nondecreasing on the half-line[0,+∞)such that for anyψ0∈Hnthe inequality
ψm(·, t) n≤βn
ψm
0n
(4.7)
takes place for allt∈Iand for allm=1,2,3, . . .(hereψmis the solution of the problem (4.2)).
Proof. First of all, one can easily verify that the functionals
Q0
ψm(·, t)=1
2 A
0
ψm(·, t)2dx,
Q1
ψm(·, t)=
A
0 1
2Dxψ
m(·, t)2 +κ
2ψ
m(·, t)4 dx,
(4.8)
are conservation laws for the problem (4.2), that is, they are independent of t. In addition, they are continuous functionals inHnbounded on bounded subsets of this
space. Hence, as above, due to the known embedding theorem for functionsg∈H1
gLp≤Cg
1/2+1/pg
x+g
1/2−1/p
, (4.9)
wherep≥2, we have
Q1ψm(·, t)+Q0ψm(·, t) ≥1
2ψ
m(·, t)2
1−Cψm(·, t)3
Dxψm(·, t)+ψm(·, t).
(4.10)
Hence,
ψm(·, t)
1≤β1ψm01
(4.11)
for allt∈Rwhereβ1(s)is a function satisfying the properties indicated above for the functionβn.
Further, solutions of the problem (4.2) satisfy the following integral equation similar to (3.1):
ψm(·, t)=e−i(t−t0)∆Pmψ0−2iκ
t
t0
e−i(t−s)∆P
mψm(·, s)2ψm(·, s)
ds. (4.12)
Fork=2, . . . , nwe get from (4.12)
ψm(·, t)
k≤C1ψm0k+C2
max
t∈I
ψm(·, t) k−1
t
t0
ψm(·, s)
kds. (4.13)
This inequality and (4.11) step by step lead to the statement ofLemma 4.1.
Proposition4.2. Letψ0∈Hn, wherenis a positive integer, and letT >0be an
arbitrary number. Then,
lim
m→∞t∈[t0max−T ,t0+T ]
ψm(·, t)−ψ(·, t)
n=0 asm → ∞, (4.14)
Proof. We get from (3.1), (4.12), and Lemmas3.2and4.1
ψm(·, t)−ψ(·, t) n
≤C1ψ0−Pmψ0n+C2 t0+T
t0
Pmψ(·, s)
2
ψ(·, s)−ψ(·, s)2ψ(·, s)nds
+C3ψ0l
t t0
ψm(·, s)−ψ(·, s) nds,
(4.15)
wherel=n−1 ifn >1 andl=1 ifn=1. The first term from the right-hand side of this inequality obviously tends to zero asm→ ∞. The integrand in the second term tends to zero for allsand is uniformly bounded insandmaccording toLemma 3.2. Therefore, this second term tends to zero, too, asm→ ∞. Thus, the statement of Proposition 4.2follows from this inequality.
Corollary4.3. For any positive integern,ψ0∈Hnandt∈R
lim
m→∞
Qkψm(·, t)−Qkψ(·, t)=0 fork=0, n. (4.16)
Proof. The proof follows from the continuity of functionalsQk,k=0, n, in the
spaceHnand the provedProposition 4.2.
As it is already noted, Lemmas4.4and4.5andProposition 4.7below are results of this section crucial for us. To establish them, we consider quantitiesQn+1(ψm(·, t)). In this connection, it should be remarked that, forψ0∈Hn, the expressionQn+1(ψ(·, t)) is not determined in general. However, since the functionsψmare infinitely
differen-tiable, the expressionQn+1(ψm(·, t))is well defined for eachm. Moreover, Lemmas 4.4and4.5andProposition 4.7take place.
Lemma4.4. Letψ0∈Hnfor a positive integernandt∈R. Then,
lim
m→∞ dQ
n+1ψm(·, t)
dt =0. (4.17)
Proof. First of all, obviouslyPmg∈C∞for any positive integermandg∈Hn.
Then, since Qn+1is a conservation law for infinitely differentiable solutions of the problem (1.1), (1.2), and (1.3), substitutingiψm
xx−2iκ|ψm|2ψminto the expression for dQn+1(ψm(·, t))/dtin place of∂ψm/∂t, we get zero. Therefore, to get the right ex-pression fordQn+1(ψm(·, t))/dt, we can bring the derivative with respect totinto the integrand and substitute the expression 2iκPm⊥[|ψm(·, t)|2ψm(·, t)]for∂ψm(·, t)/∂t
(herePm⊥=I−PmwhereIis the identity).
Using the representations (2.9) and (3.6) and the above arguments, we easily derive the following estimate:
dQn+1ψm(·, t) dt
≤γn+1ψm(·, t)nPm⊥ψm(·, t)
2
ψm(·, t)
wheren≥1 andγn+1(s)is a continuous nondecreasing function ofs∈[0,+∞). Since according toProposition 4.2and embedding theorems
Pm⊥ψm(·, t)
2
ψm(·, t) n
≤ψm(·, t)2ψm(·, t)−ψ(·, t)2ψ(·, t)n+Pm⊥ψ(·, t)
2
ψ(·, t)n →0
(4.19)
asm→ ∞, (4.18) yields the statement ofLemma 4.4.
Lemma4.5. For any positive integer nandt∈Rthere exists a functionηn+1(s),
continuous and monotonically nondecreasing on the half-lines∈[0,+∞), such that
Qn+1
ψm(·, t)−Qn+1ψ0m≤ηn+1ψ0mn
(4.20)
for all positive integermandψ0∈Hn.
Proof. The proof follows from (4.18),Lemma 4.1, and the following inequality:
Qn+1ψm(·, t)−Qn+1ψm0≤ t
t0
dQn+1ψm(·, s) ds
ds. (4.21)
Proposition4.6. For any ψ0∈Hn, wherenis a positive integer, any >0and
t∈Rthere existsδ >0such that
ψm(·, t)−ψm
1(·, t)n< (4.22)
for eachm=1,2,3, . . .and for an arbitrary solutionψm1(·, t)of problem (4.2) satisfying the condition
ψm·, t
0
−ψm
1
·, t0n< δ. (4.23)
Proof. We have from (4.12) andLemma 4.1
ψm(·, t)−ψm
1(·, t)n
≤C1δ+C2ψm
·, t0l
×
t
t0
ψm(·, s)−ψm
1(·, s)nds,
(4.24)
wherel=n−1 forn≥2 andl=1 ifn=1. This inequality step by step leads to the statement ofProposition 4.6.
The following statement together withLemma 4.5is used in the next section for provingLemma 5.5which is of principal importance for our proof ofTheorem 2.4.
Proposition4.7. LetK⊂Hnbe a compact set wherenis a positive integer. Then,
for anyt∈R
Qn+1ψm(·, t)−Qn+1ψm0
→0 asm → ∞ (4.25)
uniformly inψ0∈K(hereψm0 =Pmψ0andψm(·, t)is the solution of the problem (4.2) withψm(·, t
Proof. First of all, we prove that for any >0 andg∈Kthere existδ >0 and a
numberm0>0 such that
Qn+1ψm(·, t)−Qn+1ψm0< (4.26)
for allm≥m0andψ0∈Bδ(g)= {h∈Hn| h−gn< δ}. From (4.18) andLemma 4.1
we get
Qn+1ψm(·, t)−Qn+1ψm·, t0
≤ t
t0
γn+1βn
gn+δ
×Pm⊥ψm(·, s)
2
ψm(·, s)nds.
(4.27)
We estimate the integral from the right-hand side of (4.27). We have
Pm⊥ψm(·, s)2
ψm(·, s)
n≤ψm(·, s)
2
ψm(·, s)−ψ¯m(·, s)2¯ ψm(·, s)
n
+ψ¯m(·, s)2ψ¯m(·, s)−ψ(¯ ·, s)2ψ(¯ ·, s) n
+Pm⊥ψ(¯ ·, s)
2 ¯
ψ(·, s)n,
(4.28)
where ¯ψm and ¯ψ are the solutions of the problems (4.2) and (1.1), (1.2), and (1.3),
respectively, withψ0=g. The second and third terms in the right-hand side of this inequality are independent ofψ0∈Bδ(g)and tend to zero asm→ ∞. Further,
accord-ing toProposition 4.6, for any >0 andt∈Rthere existsδ >0 such that the first term in the right-hand side of this inequality is smaller than/2 ifψ0∈Bδ(g), for
allm. Hence, we have proved that the expression under the integral in the right-hand side of the inequality (4.27) for any fixeds becomes arbitrary small for sufficiently smallδ >0 and sufficiently large numbersmifψ0∈Bδ(g).
Further, take into accountLemma 4.1according to which the expression
P⊥
mψm(·, s)
2
ψm(·, s)n (4.29)
is bounded uniformly with respect toψ0∈K,m=1,2,3, . . . ,ands∈[t0, t]. Therefore, for anyg∈Kand >0 there existδ >0 and a numberm0such that the right-hand side of (4.27) is smaller thanifψ0∈Bδ(g)andm≥m0. So, we fix an arbitrary >0
and for anyg∈Ktakeδ=δ(g) >0 andm0=m0(g) >0 such that Qn+1
ψm(·, t)−Qn+1ψm0< (4.30)
ifψ0∈Bδ(g)(g)andm≥m0(g)(δandm0exist according to the above arguments). Then, sinceKis a compact set, there exists its finite covering by ballsBδ(g1)(g1), . . . , Bδ(gl)(gl). Letm1=max{m0(g1), . . . , m0(gl)}. Then, obviously, (4.30) is valid for all ψ0∈Kifm≥m1. Thus,Proposition 4.7is proved.
5. Proof of Theorem2.4. There is an analogy between the proof of the invariance of the measuresµnpresented below and proofs of the invariance of measures
between our construction here and the approaches in the mentioned papers consists, in particular, in the fact that, unlike in [25,26,27], in the present paper, generally speaking, the finite-dimensional measuresµm (see below) are not invariant for the
corresponding finite-dimensional dynamical systems. All the information from the general measure theory to be used further is contained in [13], for example. Then, we remark that all the results from Section 4 are valid for problem (4.3) and (4.4) with the corresponding reformulation. We fix an arbitrary positive integer n. Let g1=(e0,0),g2=(0, e0),g3=(e1,0),g4=(0, e1), . . . , g2k+1=(ek,0),g2k+2=(0, ek), . . . .
Then,{gk}k=1,2,3,...is the orthonormal basis consisting of the eigenfunctions of the
operatorSnin the spaceXn. LetLm=span{g1, . . . , g4m+2}be the sequence of
finite-dimensional subspaces of the spaceXn(m=1,2,3, . . .). In the spacesLmwe introduce
the following finite-dimensional Gaussian measures:
wm(Ω)=(2π )−(2m+1)
2m
k=0 ωk
F
e−(1/2)k2m=0ωk(pk2+q2k)dp0dq0···dp2mdq2m, (5.1)
whereΩ= {u∈Lm|[(u, g1)n, . . . , (u, g4m+2)n]∈F}andF⊂R4m+2is a Borel set. So, wmis a centered Gaussian measure in the spaceLmfor any positive integerm. We
recall that the spaceLmis equipped with the topology generated by the norm of the
spaceXn.
Further, letwm(Ω)=wm(Ω∩Lm)for any Borel subsetΩof the spaceXn. We state
that, with this definition, the measurewm becomes a well-defined Borel measure in
the spaceXn. To prove this statement, it suffices to show thatΩ∩Lmis a Borel subset
of the spaceLm for any Borel setΩ⊂Xn. Indeed, it is clear that the set of all
subsets of the spaceLm of this kind is a sigma-algebra of subsets of the spaceLm
which obviously contains all open subsets of this space. Therefore, it suffices to prove thatis the minimal sigma-algebra of subsets of the spaceLm containing all open
subsets of this space. Suppose that this is not right. Then, the Borel sigma-algebra1 of the spaceLmis contained in the sigma-algebraand1=. Consider the setᏺ
of all Borel subsetsAof the spaceXnsuch thatA∩Lm∈1. Then, it is clear thatᏺ
is a sigma-algebra in the spaceXnwhich contains all open subsets of the spaceXn
and is more narrow than the Borel sigma-algebra of this space. Thus, we arrive at a contradiction and, therefore, we have proved thatΩ∩Lmis a Borel subset of the space Lmfor an arbitrary Borel subsetΩof the spaceXn.
Lemma5.1. The sequence{wm}m=1,2,3,...of the Borel measures weakly converges to
the measurewnin the spaceX n.
Remark5.2. We recall that, by definition, a sequence of bounded Borel measures
{νk}k=1,2,3,...weakly converges to a bounded Borel measureνin a separable complete
metric spaceHif
lim
k→∞
H
Φ(g)dνk(g)=
H
Φ(g)dν(g) (5.2)
for an arbitrary real continuous bounded functionalΦin the spaceH.
Remark5.3. A statement similar toLemma 5.1is presented in [27]; here we make
Proof ofLemma5.1. As it is proved in [27], the sequence of measures{wm}is
weakly compact in the spaceXn. In addition, for any cylindrical setM⊂Xnwe have wm(M)=wn(M)for sufficiently large numbersm.
Suppose that the sequence {wm}does not converge weakly to the measurewn.
Then, there exists a Borel measurewinXn, not coinciding with the measurewn, and
a subsequence{wmk}of the sequence{wm}such that the sequence{wmk}weakly converges to the measurew. Further, since the extension of a measure, countably additive on an algebra, to the minimal sigma-algebra containing this algebra is unique (see [13]), there exists a cylindrical set of the kind
M=g∈Xn|
g, g1
n, . . . ,
g, g4r+2n
∈F, (5.3)
whereris a positive integer andF⊂R4r+2is a Borel set, such thatw(M)≠wn(M).
Then, due to the known property of Borel sets, there exists a sequenceF1⊃F2⊃ ··· ⊃ Fi⊃ ··· of open setsFi⊂R4r+2such thatF=
i≥1Fi. Therefore,
w(M)=lim
i→∞w M
i
, wn(M)=lim i→∞w
nM i
, (5.4)
where
Mi=g∈Xn|g, g1n, . . . ,
g, g4r+2n
∈Fi. (5.5)
Hence, we can accept that the cylindrical setM⊂Xnis open.
Take an arbitrary >0 and consider a continuous functionalΦ(z)=Φ(z1g1+ ···+z4r+2g4r+2)depending on variablesz=(z1, . . . , z4r+2)∈R4r+2and satisfying the following properties:
(1) 0≤Φ(z)≤1 for allz;
(2) Φ(z)=0 ifg=z1g1+···+z4r+2g4r+2∈M;
(3) Φ(z)=1 ifg∈Mand dist(g;∂M)≥where for an arbitrary setA⊂Xnand g∈Xndist(g;A)=infh∈Ag−hn.
It is clear that
Xn
Φ(z)dwm(g)=
Xn
Φ(z) dwn(g) (5.6)
form≥r. Further, lim→+0XnΦ(z) dw
n(g)=wn(M). At the same time, there exists
>0 such that each of the expressions
wn(M)−
Xn
Φ(z) dwn(g)
,
w(M)−lim
k→∞
Xn
Φ(z) dwmk(g)
=w(M)−
Xn
Φ(z) dw(g)
(5.7)
is smaller than|w(M)−wn(M)|/3. This fact contradicts (5.6). Thus,Lemma 5.1is
proved.
We introduce the following measures:
µm(Ω)=
Ωe
Φn(u,v)dw
(hereΩ⊂Xnare Borel sets). Since due to the representations (2.9) and (3.6) the
func-tionalΦn(u, v)is continuous inXnand bounded on any bounded subset of the space Xn, the measuresµmandµnare well defined inXn.
With the following statement, we in fact establish a well-known property of weakly converging sequences of measures.
Lemma5.4. One haslim infm→∞µm(Ω)≥µn(Ω)for any open bounded setΩ⊂Xn
andlim supm→∞µm(K)≤µn(K)for any closed bounded setK⊂Xn.
For the proof, see, for example, [27].
Letfm(u0, v0, t)=(um(·, t+t0), vm(·, t+t0))for any(u0, v0)∈Xnandt∈Rwhere (um(·, t), vm(·, t))is the solution of problem (4.3) and (4.4). One of the principal
differ-ences between the constructions in the present paper and in earlier papers [25,26,27] is that, unlike in [25,26,27], in the present case the finite-dimensional measuresµm
are not in general invariant for the corresponding DSfm. In this connection, we note
that the following statement is crucial for us; it shows that the measuresµmare
“al-most invariant” asymptotically for sufficiently large numbersm.
Lemma5.5. Lett∈RandΩ⊂Xnbe an arbitrary closed bounded set. Then,
lim
m→∞
µm
fm(Ω, t)
−µm(Ω)
=0. (5.9)
Proof. We rewrite system (4.3) for coordinatesp(t)=(p0(t), . . . , p2m(t))andq(t)=
(q0(t), . . . , q2m(t))whereum(x, t)=2km=0pk(t)ek(x)andvm(x, t)=2km=0qk(t)ek(x).
Then, we get the following system of ordinary differential equations
˙ pk(t)=
∂h(p, q) ∂qk
1+λnk
, k=0,2m,
˙ qk(t)= −
∂h(p, q) ∂pk
1+λnk, k=0,2m.
(5.10)
Here h(p, q)= Q1(um, vm) = (1/2) A
0{(umx)2+(vxm)2+κ((um)2+(vm)2)2} dx.
Therefore, according to the well-known result [19,27], the Lebesgue measureσmin the
phase spaceR4m+2is an IM for this DS (in fact, this is a well-known statement accord-ing to which the Lebesgue measure defined in the phase space of a finite-dimensional Hamiltonian system is an IM for this system). This easily implies that
D=det ∂
p(t), q(t) ∂p(0), q(0)
≡1 (5.11)
for an arbitrary solution(p(t), q(t))of (5.10).
Fix an arbitrary bounded closed set Ω⊂Xn. Then, according to the above
argu-ments, we have
µm
fm(Ω, t)
=
Ω
eQn+1(Pmw)−Qn+1(fm(w,t))dµ
m(w). (5.12)
Further,
µm(Ω)−µm
fm(Ω, t)≤
Ω
1−eQn+1(Pmw)−Qn+1(fm(w,t))dµ
According to Lemma 4.5, the integrand here is uniformly bounded with respect to w∈Ωby a constantC >0. Fix an arbitrary >0. Then, according to the Prokhorov’s theorem (see [10]), there exists a compact setK⊂Ωsuch thatµm(Ω\K) < for all m=1,2,3, . . .andµn(Ω\K) < . Further, in view ofProposition 4.7and (5.13),
lim
m→∞
µm(K)−µm
fm(K, t)
=0, (5.14)
therefore
lim sup
m→∞
µm(Ω)−µm
fm(Ω, t)≤C, (5.15)
whereC >0 depends only on supg∈Ωgn. Hence, due to the arbitrariness of >0, Lemma 5.5is proved.
Corollary5.6. For any bounded open setΩ⊂Xnandt∈R,
lim
m→∞
µm(Ω)−µmfm(Ω, t)=0. (5.16)
The further proof ofTheorem 2.4includingLemma 5.7below in fact repeats a con-struction in [25,26,27].
Lemma 5.7. Let Ω ⊂ Xn be an open bounded set and t ∈ R. Then, µn(Ω) =
µn(fn(Ω, t)).
Proof. According toTheorem 2.3 and Lemma 3.2, fn(Ω, t) is a bounded open
subset of the spaceXn. Fix an arbitrary >0. Then, sinceµn is a Radon measure,
there exists a compact setK⊂Ωsuch thatµn(Ω\K) < . LetK
1=fn(K, t). Then,K1 is a compact set, too, andK1⊂fn(Ω, t)=Ω1. Letα=min{dist(K, ∂Ω); dist(K1, ∂Ω1)} where dist(A, B)=infu∈A;v∈Bu−vn for anyA, B⊂Xnand ∂Ais the boundary of
the setA. Obviouslyα >0. Due toProposition 4.6for anyg∈Kthere exists a ball Br(g)= {h∈Xn| g−hn< r} ⊂Ωwith a positive radiusrsuch that
fm(g, t)−fm(h, t)n< α
3 (5.17)
for allh∈Br(g)and for allm. LetB1, . . . , Blbe a finite covering of the compact set Kby these balls. LetΩβ1= {h∈Ω1|dist(h, ∂Ω1)≥β}whereβ >0, andD=
l i=1Bi.
Since in view ofProposition 4.2fm(g, t)→fn(g, t)inXnasm→ ∞for anyg∈Xn,
we get
fm(D, t)⊂Ω1α/4 (5.18)
for all sufficiently large numbersm.
Further, according to Lemmas5.4,5.5, andCorollary 5.6
µn(Ω)≤µn(D)+≤lim inf
m→∞ µm(D)+ =lim inf
m→∞ µm
fm(D, t)
+
≤µnΩα/4 1
+≤µnΩ
1
+.
Therefore, due to the arbitrariness of >0, we haveµn(Ω)≤µn(Ω
1). By analogy, µn(Ω)≥µn(Ω
1). Hence,µn(Ω)=µn(Ω1)andLemma 5.7is proved.
We proveTheorem 2.4. Let firstΩ⊂Xnbe an open (unbounded) set. LetΩk= {w∈
Ω| fn(w, t)
n+wn< k}. Then,Ω=
∞
k=1Ωkand each setΩkis bounded and open. In addition,fn(Ω, t)=∞
k=1fn(Ωk, t)andµn(Ωk)=µn(fn(Ωk, t)). Therefore,
µnfn(Ω, t)=lim
k→∞µ
nfnΩk, t=lim k→∞µ
nΩk=µn(Ω). (5.20)
Further, for an arbitrary Borel subsetAof the spaceXnwe get the equalityµn(A)= µn(fn(A, t)) approximating this setA from outside by open sets (we remark that,
since the functionfn(·, t)is a homeomorphism for any fixedt, it transforms Borel
sets into Borel ones). The last statement of Theorem 2.4 follows from Lemma 3.2. Theorem 2.4is proved.
Remark 5.8. As it was noted in the introduction, an IM for the NLS generated
by the conservation lawQ1is constructed in [5], for example. Here, we presented IMs generated by the conservation lawsQnwheren≥2. The question is left open whether
there is an IM associated with the conservation lawQ0. As one could observe from our considerations, to any conservation lawQn (n≥2)there corresponds an IM on the
phase spaceHn−1. Therefore, it seems to be probable that such an IM could exist on a phase space likeH−1. However, since the well-posedness of the problem (1.1), (1.2), and (1.3) in a similar sense seems to be open and very difficult, we could not make a construction like that.
Acknowledgement. The author is thankful to Dr A. P. Isaev for useful
discus-sions and to Mrs T. Dumbrajs for editing the text.
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for Nuclear Research,141980Dubna (Moscow Region), Russia
