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Applicable AnalysisVol. 00, No. 00, Month 200x, 1–10

RESEARCH ARTICLE

Global Well-Posedness for Discrete Nonlinear Schrodinger

Equation

Gaston M. N’Guerekata and Alexander PankovDepartment of Mathematics

Morgan State University1700 E. Cold Spring Lane

Baltimore, MD 21251, USA(Received 00 Month 200x; in final form 00 Month 200x)

We consider the initial value problem for the discrete nonlinear Schrodinger equation inweighted l2-spaces. Under quite general assumptions we prove that the problem is globallywell-posed. The proof is based on simple general facts on abstract evolution equations.

Keywords: global well-posedness, discrete Schrodinger equation, mild solution, semigroupsof operators

AMS Subject Classification (2000): 35Q55, 35A05

1. Introduction

In this paper we consider the initial value problem for the time dependent discretenonlinear Schrodinger equation (DNLS):

iun = −∆un + vnun − fn(un) , n ∈ Z , (1)

un(0) = u0n . (2)

Here ∆ stands for the discrete one dimensional Laplacian

∆un = un+1 + un−1 − 2un . (3)

This equation, as well as its version for multi-dimensional spatial lattice, appears inmany applications (see, e.g., [5]). We deal with the case of one-dimensional spatiallattice for simplicity of the notation. All the results extend straightforwardly tomulti-dimensional case.

Our starting point is paper [7] by P. Pacciani, V. V. Konotop and G. PerlaMenzala. In that paper the authors study the simplest case equation (1) whenvn = 0 and fn(u) = χ|u|p−1u, p > 1, is the power nonlinearity independent of n.Those authors prove global well-posedness of the initial value problem in spacesof power decaying data. This is in the strike contrast with the case of continuousnonlinear Schrodinger equation. As it is well-known, in the latter case finite timeblow-up occurs provided the nonlinearity growths faster than certain critical power(see, e.g., [2, 14]). For instance, this is so for one dimensional continuum NLS withpower nonlinearity if p ≥ 5. The techniques used in [7] is based on the explicitrepresentation of the Green function of the operator id/dt−∆.

ISSN: 0003-6811 print/ISSN 1563-504X onlinec© 200x Taylor & FrancisDOI: 10.1080/0003681YYxxxxxxxxhttp://www.informaworld.com

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The well-posedness result of [7] is not quite satisfactory because of, at least, tworeasons. First, it is known that in many cases DNLS (1) possesses standing wave so-lutions with exponentially decaying amplitude (see [9, 10, 15]). Therefore, it wouldbe interesting to have a global well-posedness result in spaces of exponentially de-caying data as well. The second point is that paper [7] only deals with translationinvariant DNLS though the non-translation invariant DNLS is not less important.Applications of DNLS range from nonlinear optics and condensed matter physicsto biology, and translation invariance is just the simplest model assumption. Let usalso point out that DNLS considered in [9] and [15] are not translation invariant.

In this paper we prove that the global well-posedness for problem (1), (2) takesplace in a wide class of weighted spaces under minimum assumptions on the poten-tial and nonlinearity. The proof is based on elementary facts on abstract semilineardifferential equations, conservation of charge (or number of particles), and a simplecut-off argument. Similar approach is used in [1] to obtain global well-posednessresults for equations of infinite systems of coupled nonlinear oscillators. However,in the contrast to the case of DNLS, in certain systems of oscillators blow-up mayoccur.

The organization of the paper is the following. In Section 2 we remind somepreliminaries about abstract differential equations. We formulate the main result,Theorem 3.1, in Section 3, while Section 4 contains the proof. Finally, in Section 5we discuss some further extensions of Theorem 3.1.

2. Preliminaries

We treat equation (1) as an abstract differential equation of the form

u′ = Au+N(u) (4)

in a complex Banach (actually, Hilbert) space. We always assume that A is a closed(in general unbounded) operator in a Banach space E, with the domain D(A), andN is a continuous operator from E into itself.

Let us remind certain elementary facts related to such equation (see, e.g., [4,12, 13]). A family U(t), t ∈ [0,∞), of bounded linear operators in E is a stronglycontinuous semigroup of operators if

(a) U(t)v is a continuous function on [0,∞) with values in E for every v ∈ E;(b) U(0) = I — the identity operator in E;(c) U(t+ s) = U(t)U(s) for all t, s ∈ [0,∞).

If the family U(t) is defined for all t ∈ R and satisfies (i)–(iii) on whole real line,we say that U(t) is a strongly continuous group of operators.

If U(t) is a strongly continuous semigroup of operators, then its generator A isdefined by

Av = limt→0+

t−1(U(t)− I)v , (5)

with the domain D(A) consisting of those v ∈ E for which the limit in (5) exists. IfU(t) is a group, then the one-sided limit in (5) can be replaced by two-sided limit.

Proposition 2.1 (a) If A is a generator of a strongly continuous semigroup in aBanach space E and B is a bounded linear operator in E, then A+B is a generatorof a strongly continuous semigroup.

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(b) If H is a self-adjoint operator in a Hilbert space, then A = iH is a generatorof a strongly continuous group.

If A is a bounded linear operator, then it generates a one parameter group etA.In general, if A is a generator of a strongly continuous semigroup, we still use thesame exponential notation etA for the semigroup generated by A.

Now we discuss the initial value problem for equation (4), with initial data

u(0) = u0 ∈ E . (6)

If A is a bounded operator, then it is enough to consider classical solutions, i.e.,continuously differentiable functions with values in E that satisfy (4), (6). In gen-eral, when the operator A is unbounded, we consider mild solutions to (4), (6). Acontinuous function u on [0, T ] with values in E is a mild solution of the initialvalue problem (4), (6) if it satisfies the following integral equation

u(t) = etAu0 +∫ t

0e(t−s)AN(u(s)) ds . (7)

In the case when the operator A is bounded, these are classical solutions.

Our main tool in the following Section is the following well-known result (cf. [11]Theorems 1.2 and 1.4 pp. 184-186).

Proposition 2.2 Let A be a generator of a strongly continuous semigroup in aBanach space E and N be a locally Lipschitz continuous operator in E, i.e., forevery R > 0 there exists C = C(R) > 0 such that

‖N(w)−N(w′)‖ ≤ C‖w − w′‖ (8)

whenever ‖w‖ ≤ R and ‖w′‖ ≤ R. Then for every u0 ∈ E there exists a uniquelocal solution of initial value problem (4), (6) defined on certain interval [0, τ).

If in addition N is globally Lipschitz continuous, i.e., there exists a constantC > 0 such that

‖N(w)−N(w′)‖ ≤ C‖w − w′‖ ∀w,w′ ∈ E , (9)

then problem (4), (6) possesses a unique global solution defined on [0,∞). More-over, the solution u(t) depends continuously on u0 in the topology of uniformconvergence on bounded subintervals of [0,∞).

Remark 1 : Assumption (8) implies automatically that N is bounded onbounded sets.

3. Main Result

We consider equation (1) under the following assumptions.

(i) The potential V = (vn) is a sequence of real numbers.(ii) The nonlinearity fn : C→ C satisfies

fn(eiωz) = eiωfn(z)

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(gauge invariance), fn(0) = 0, fn(z) = o(|z|) as z → 0, and is uniformlylocally Lipschitz continuous: for every R > 0 there is a constant C = C(R)independent of n ∈ Z such that

|fn(z)− fn(z′)| ≤ C|z − z′|

for all n ∈ Z whenever |z| ≤ R and |z′| ≤ R.

The function fn can be represented in the form

fn(z) = gn(|z|2)z , (10)

where gn : [0,∞)→ R. Examples of such nonlinearities are power nonlinearity

fn(z) = γn|z|p−1z , p > 1 , γn ∈ R ,

and saturable nonlinearities such as

fn(z) = γn|z|p−1z

1 + |z|p−1, p > 1 , γn ∈ R ,

and

fn(z) = γn(1− exp(−an|z|2))z , an > 0 , γn ∈ R .

The space l2 consists of two-sided infinite sequences u = (un) of complex numberssuch that the norm

‖v‖ = ‖v‖l2 = (∑n∈Z|vn|2)1/2

is finite. The space l2 is a Hilbert space with the inner product

(u, v) =∑Z

unvn .

The space l∞ consists of all bounded two-sided sequences. Endowed with the norm

‖u‖l∞ = supZ

|un| ,

this is a Banach space, and l2 is densely imbedded into l∞, with

‖u‖l∞ ≤ ‖u‖l2 . (11)

We are looking for solutions in weighted l2-spaces. Let Θ = (θn) be a sequenceof positive numbers (weight). The space l2Θ consists of all two-sided sequences ofcomplex numbers such that the norm

‖u‖Θ = (∑n∈Z|un|2θn)1/2

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is finite. This is a Hilbert space, with the inner product

(u,w)Θ =∑n∈Z

unwnθn ,

continuously and densely embedded into l2.We always suppose that the weight Θ satisfies the following regularity assump-

tion:

(iii) the sequence Θ is bounded below by a positive constant and there exists aconstant c0 > 0 such that

c−10 ≤ θn+1

θn≤ c0

for all n ∈ Z.

Obviously, l2Θ is densely and continuously embedded into l2. If θn ≡ 1, thenl2Θ = l2. From the point of view of functional analysis assumption (iii) is quitenatural. It means that the space l2Θ is translation invariant. More precisely, let T+

and T− be the operators of right and left shifts, respectively, defined by

(T+w)n = wn−1 and (T−w)n = wn+1 .

Assumption (iii) holds if and only if both T+ and T− are linear bounded operatorsin l2Θ. Note that T+ and T− are mutually inverse operators. But let us point outthat the translation invariance of the space l2Θ does not mean that the norm ‖ · ‖l2Θis translation invariant.

The most important examples of regular weights are

(a) power weight

θn = (1 + |n|)b , b > 0 ; (12)

(b) exponential weight

θn = exp(α|n|) , α > 0 . (13)

More generally, the weight θn = exp(α|n|β), α > 0, satisfies assumption (iii) if andonly if 0 < β ≤ 1.

Our main result is the following

Theorem 3.1 : Under assumptions (i)–(iii), for every u0 ∈ l2Θ problem (1), (2)has a unique global solution u ∈ C(R, l2Θ). Furthermore, we have that

‖u(t)‖2 = ‖u0‖2 , t ∈ R . (14)

Remark 1 : The conserved quantity ‖u(t)‖2 usually is referred to as a charge ornumber of particles.

4. Proof of Main Result

We need several remarks and lemmas before presenting the proof of the theorem.

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First, let us note that formally, equation (1) is of the form (4), where A = −iH,

H = −∆ + V (15)

and the operator N is defined by

N(u)n = ifn(un) . (16)

Let us consider certain properties of the operator H as an operator in the spacel2Θ (in general, unbounded). The operator (of multiplication by) V is a diagonaloperator and, hence, a self-adjoint operator in l2Θ with the domain

DΘ = {u ∈ l2Θ : V u ∈ l2Θ} . (17)

We drop the index Θ and denote by D the domain of the operator V in l2. Obvi-ously, DΘ ⊂ D.

The operator −∆ is a bounded operator in l2Θ. Indeed, −∆ can be representedas

−∆ = −∂−∂+ ,

where

∂−un = un−1 − un , ∂+un = un+1 − un .

Now we have

‖∂+u‖2Θ =∑n∈Z|un+1 − un|2θn ≤ 2(

∑n∈Z|un+1|2θn +

∑n∈Z|un|2θn) .

By (iii), this implies that

‖∂+u‖2Θ ≤ 2(c0 + 1)∑n∈Z|un|2θn) = 2(c0 + 1)‖u‖2Θ .

Similarly,

‖∂−u‖2Θ ≤ 2(c0 + 1)‖u‖2Θ .

Thus, both ∂+ and ∂− are bounded operators in l2Θ, hence so is −∆. Furthermore,it is easily seen that in l2 the operators ∂+ and ∂− are mutually skew-adjoint,

(∂+)∗ = −∂− ,

and commute. This implies immediately that −∆ is a bounded self-adjoint operatorin l2.

The previous remarks and Proposition 2.1 imply the following

Lemma 4.1: The operator −iH, with the domain DΘ, is a generator of stronglycontinuous group exp(−itH) in the space l2Θ. In l2 this is a unitary group.

The basic property of the operator N is given in

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Lemma 4.2: Under assumption (ii) the operator N defined by (16) acts inl2Θ as a locally Lipschitz continuous operator (see (8) in Proposition 2.2). If thenonlinearity fn is uniformly Lipschitz continuous, i.e., there is a constant C > 0,independent of n, such that

|fn(z)− fn(z′)| ≤ C|z − z′| ∀z, z′ ∈ C , (18)

then the operator N is Lipschitz continuous in l2Θ:

‖N(w)−N(w′)‖Θ ≤ C‖w − w′‖Θ ∀w,w′ ∈ l2Θ .

Proof . Suppose that ‖w‖Θ ≤ R and ‖w′‖Θ ≤ R. Due to the continuous embed-ding l2Θ ⊂ l2, we see that ‖w‖ ≤ R′ and ‖w′‖ ≤ R′, with some R′ > 0. Hence, byinequality (11), ‖w‖l∞ ≤ R′ and ‖w′‖l∞ ≤ R′. Now assumption (ii) implies that

‖N(w)−N(w′)‖2Θ =∑n∈Z|fn(wn)− fn(w′n)|2θn ≤

≤ C(R′)∑n∈Z|wn − w′n|2θn = C(R′)‖w − w′‖2Θ .

The second statement of the lemma is trivial.�

Remark 1 : It is easily seen that any mild solution of (1) in l2Θ is a mild solutionin l2.

Proof of Theorem 3.1. First, we consider problem (1), (2) under an additionalassumption that the nonlinearity is uniformly Lipschitz continuous (see (18)). Inthis case the existence of a unique global solution follows from Lemmas 4.1 and4.2, and Proposition 2.2.

Let us prove conservation law (14) under the same extra assumption. First,suppose that u0 ∈ D. Then the solution u(t) is actually a classical solution. Takingthe imaginary part of the inner product of equation (1) and making use of gaugeinvariance (namely, equation (10)), we obtain that

12

(‖u‖2)′ = 0 .

Hence, ‖u(t)‖2 = ‖u0‖2. If u0 ∈ l2, we can choose u0,k ∈ D such that u0,k → u0 in l2.By the last statement of Proposition 2.2, solutions uk(t) of the initial value problemwith the initial data u0,k converge to u(t) uniformly on each finite interval [−T, T ].This implies (14) for every mild solution in l2 (under the additional assumption onthe nonlinearity imposed above).

Finally, let us remove the uniform Lipschitz continuity of the nonlinearity. Letu0 ∈ l2Θ and r0 = ‖u0‖2. We introduce a function gn : [0,∞) → R such thatgn(r) = gn(r) if 0 ≤ r < r0, and gn(r) = gn(r0) if r ≥ r0, where gn is defined by(10). Next, we set

fn(z) = gn(|z|2)z .

This function is uniformly Lipschitz continuous. Hence, there exists a unique mild

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solution of equation (1), with fn replaced by fn, in l2Θ such that u(0) = u0. But

‖u(t)‖2l∞ ≤ ‖u(t)‖2 = r0 .

By the definition of fn, we see that

fn(un(t)) = fn(un(t)) .

Therefore, u is a solution of the original equation. Due to local uniqueness, thereis no other solution, and the proof is complete.

�

5. Extensions

The result of Theorem 3.1 extends easily to some more general situations. Herewe consider two such cases, namely, the DNLS with long range linear inter-siteinteractions and the DNLS with dissipation. In both cases, the results are notmost general because we would like to keep the presentation to be transparent.

The DNLS with long range interaction is of the form

iun = Kun + vnun − fn(un) , n ∈ Z , (19)

where the operator K is defined by

Kun =∞∑

m=−∞Kn−mum

and the kernel Kn is symmetric, K−n = Kn. In particular, K0 is real.

Theorem 5.1 : In addition to assumptions (i)–(iii), suppose that there exists aconstant C > 0 such that

θn+m ≤ Cθnθm (20)

for all n ∈ Z and m ∈ Z, and

∞∑n=−∞

|Kn|θn <∞ . (21)

Then for every u0 ∈ l2Θ equation (19) has a unique solution u ∈ C(R, l2Θ) such thatu(0) = u0. Moreover,

‖u(t)‖2 = ‖u0‖2

for all t ∈ R.

Assumptions (20) and (21) guarantee that the operator K as a linear boundedoperator in both l2 and l2Θ. It is easily seen that the operator K is self-adjoint inl2. Now the same arguments as in the proof of Theorem 3.1 prove Theorem 5.1.

The most interesting weights such as power and exponential weights (12) and(13), respectively, satisfy assumption (20). On the other hand, there are weaker,

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REFERENCES 9

but less convenient, assumptions than (20) and (21) under which the conclusion ofTheorem 5.1 holds true. We also mention that in the case of finite range interaction,when Kn = 0 if |n| > n0 with some integer n, the global well-posedness of the initialvalue problem in l2Θ takes place only under assumptions (i)–(iii). Extra assumption(20) is not needed in this case.

Remark 1 : Theorem 5.2 in [7] contains a gap. Assumption

∞∑n=−∞

|Kn|θn <∞

imposed in that paper is not sufficient to perform the proof because, under thisassumption, the dispersion relation ω(σ) is only continuous, in general not differ-entiable. Therefore, integration by parts used in [7] is not possible.

Now we show that dissipations effects can also be included. Consider the equation

iun = −∆un + vnun − fn(un) + iγun , n ∈ Z , (22)

where γ > 0.

Theorem 5.2 : Under assumptions (i)–(iii), for every u0 ∈ l2Θ equation (22) hasa unique solution u ∈ C([0,∞), l2Θ) such that u(0) = u0. Moreover,

‖u(t)‖2 ≤ ‖u0‖2 , t ∈ [0,∞) . (23)

The proof is similar to that of Theorem 3.1. The main difference is that now wehave, and use, inequality (23) instead of conservation of charge, and solutions areconsidered only on [0,∞).

Finally, we point out that the proofs do not use any specific feature of onedimensional lattice. Therefore, all the results extend immediately to the case ofmulti-dimensional DNLS. The only change is that the notation becomes more com-plicated.

References

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Acad. Press, New York, 1975.[14] C. Salem, P.-L. Salem, The Nonlinear Schrodinger Equation, Springer, New York, 1999.

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[15] G. Zhang, A. Pankov, Standing waves of the discrete nonlinear Schrodinger equation with growingpotential, II, Appl. Anal., to appear.