The closeness of the Ablowitz-Ladik lattice to the Discrete Nonlinear Schrödinger equation
aa r X i v : . [ n li n . PS ] F e b The closeness of the Ablowitz-Ladik lattice to the Discrete Nonlinear Schr¨odingerequation
Dirk Hennig and Nikos I. Karachalios
Department of Mathematics, Laboratory of Applied Mathematics and Mathematical Modelling,University of the Aegean, 83200, Karlovassi, Samos, Greece
Jes´us Cuevas-Maraver
Grupo de F´ısica No Lineal, Departamento de F´ısica Aplicada I,Universidad de Sevilla. Escuela Polit´ecnica Superior, C/ Virgen de ´Africa, 7, 41011-Sevilla, SpainInstituto de Matem´aticas de la Universidad de Sevilla (IMUS). EdificioCelestino Mutis. Avda. Reina Mercedes s/n, 41012-Sevilla, Spain (Dated: February 11, 2021)While the Ablowitz-Ladik lattice is integrable, the Discrete Nonlinear Schr¨odinger equation, whichis more significant for physical applications, is not. We prove closeness of the solutions of bothsystems in the sense of a “continuous dependence” on their initial data in the l and l ∞ metrics. Themost striking relevance of the analytical results is that small amplitude solutions of the Ablowitz-Ladik system persist in the Discrete Nonlinear Schr¨odinger one. It is shown that the closenessresults are also valid in higher dimensional lattices as well as for generalized nonlinearities, thelatter is exemplified by generic power-law and saturable ones. For illustration of the applicabilityof the approach a brief discussion of the results of numerical studies is included, showing that when1-soliton solution of the Ablowitz-Ladik system is initiated in the Discrete Nonlinear Schr¨odingersystem with cubic and saturable nonlinearity, its long-term persistence. Thereby excellent agreementof the numerical findings with the theoretical predictions are obtained. I. INTRODUCTION
The Discrete Nonlinear Schr¨odinger equation (DNLS) i ˙ φ n + κν ( φ n +1 + φ n − ) + γ | φ n | φ n = 0 , n ∈ Z , γ ∈ R , (1.1)is one of the most important nonlinear lattice systems [1],[2],[3],[4]. It appears as a fundamental, inherently discretemodel in a great variety of physical contexts. The study of its dynamics has been an exciting topic of research as itdeals with such diverse physical and biological phenomena as wave motion in coupled nonlinear waveguides, dynamicsof modulated waves in nonlinear electric lattices, localization of electromagnetic waves in photonic crystals, energylocalization in discrete condensed matter and biological systems and the dynamics of Bose-Einstein Condensates [5],to mention a few. In Eq. (1.1), κ > ν and γ rendersthe DNLS as focusing (same sign) or defocusing (opposite sign).A crucial difference from its continuous limit κ → ∞ leading to the cubic Nonlinear Schr¨odinger equation (NLS) i∂ t u + νu xx + γ | u | u = 0 , x ∈ R , (1.2)is that the DNLS (1.1) is a non-integrable discretization of the integrable partial differential equation (1.2). This isnot the case for another discretization of NLS (1.2), known as the Ablowitz-Ladik equation (AL) [6], [7], [8], i ˙ ψ n + κν ( ψ n +1 + ψ n − ) + µ | ψ n | ( ψ n +1 + ψ n − ) = 0 , n ∈ Z , µ ∈ R . (1.3)Like in (1.1) the sign of the parameters ν and µ determines whether the model is of focusing or defocusing kind.Most importantly, the AL lattice is an integrable discretization of (1.2), as it was shown by the discrete version of theInverse Scattering Transform [6], and therefore it has an infinite number of conserved quantities. We remark that theAL is one of the few known completely integrable (infinite) lattice systems admitting soliton solutions [9],[10]. On theinfinite lattice with vanishing boundary conditions, general solutions can be obtained [7]. For example, for κ = ν = 1and µ >
0, the one-soliton solution reads as ψ sn = sinh β √ µ sech [ β ( n − ut )] exp( − i ( ωt − αn )) ,ω = − α cosh β,u = 2 β − sin α sinh β, (1.4)with α ∈ [ − π, π ] and β ∈ [0 , ∞ ). In [11], the Inverse Scattering Transform method has been developed for nonvanishingboundary conditions and N-dark soliton solutions of the AL equation have been given in terms of the Casoratideterminant in [12]. Yet in similarity to the integrable NLS (1.2), another important class of solutions of the AL isthe one of rational solutions which are discrete versions of the Peregrine soliton and the Kuznetsov-Ma breather [13],[14].Strictly speaking, all the aforementioned analytical solutions exist only for the AL. In the case of the soliton solution(1.4) of the AL this is manifested in the fact it exhibits continuous translation symmetry and possesses a band ofvelocities for each β , which allow it to travel along the lattice. This is not the case for the DNLS for which a localizedstate can be pinned due to the Peierls-Nabarro barrier [15],[16], [17].Therefore, the question of the persistence of solitary wave dynamics and the existence of localized structures innon-integrable lattices, such as the DNLS (1.1), has attracted tremendous interest. As a milestone in this context,we recall the construction of localized in space, time periodic or time-quasiperiodic solutions of lattice dynamicalsystems, including the DNLS as special case [18], [19],[20], starting from the anti-continuous limit κ →
0. Forkey works regarding numerical computations of discrete solitons we refer to [21],[22]. Other seminal results on theexistence of nonlinear localized modes of the DNLS equations have been produced by nonlinear analysis methods, inparticular variational ones, establishing the existence of localized structures as critical points of suitable functionals[23],[24],[25],[26],[27]. Important extensions concern the existence of more complex structures in higher dimensionalset-ups, such as the discrete vortex solutions, see [4] and references therein. The crucial issue of existence and stabilityof travelling solitons in DNLS lattices has been investigated by a combination of analytical and numerical methodsverifying in many cases the robustness of the discrete localized structures under perturbations, [28],[29],[30],[31]. Inthis context we also refer to the reviews [32], [33], [34].In the present work, we investigate a persistence/existence problem, by examining in the sense of “continuousdependence”, the closeness of the solutions of the DNLS and the AL for close enough initial data. That is, thefollowing question is investigated in the present paper: assuming that the initial data of the DNLS (1.1) and the AL (1.3) are sufficiently close in a suitable metric, do the associated solutions remain close for sufficiently long times?
We argue that answering this question is important because of the following reasons: • Whereas, it is natural to expect, at least in some cases of parametric regimes, that sufficiently weak non-integrable perturbations (e.g. stemming from gain/loss or forcing terms or higher order terms) lead to solutionsstaying close to those of the underlying integrable (core) system dynamics, in the case of the DNLS and ALlattices, there is not such a limiting connection between the systems. • An affirmative answer to the question above will establish that the already diverse dynamical features of theDNLS itself are even further enriched as then the DNLS closely share such solutions that are provided bythe functional form of the analytical solutions of the AL-lattice (at least) for small amplitudes. From thisperspective, not only the soliton solutions but also the discrete rational solutions are relevant.In our aim to answer the above question, we proceed by analytically proving that at least under certain smallnessconditions on the initial data of the DNLS and the AL lattices, the corresponding solutions remain close for all times.
To be precise, we state the result for the infinite lattice with vanishing boundary conditionslim | n |→∞ φ n = lim | n |→∞ ψ n = 0 . (1.5)Hence, the natural phase space for the systems is the Hilbert-space of the square-summable sequences l = φ = ( φ n ) n ∈ Z ∈ C | || φ || l = X n ∈ Z | φ n | p ! / . (1.6)Consider then, the initial conditions for the DNLS (1.1) and AL (1.3) φ n (0) = φ n, , n ∈ Z , (1.7)and ψ n (0) = ψ n, , n ∈ Z , (1.8)respectively. The first of the main results of the paper is the following Theorem I.1.
Consider the DNLS equation (1.1) . We assume that for every ǫ > , the initial conditions (1.7) ofthe DNLS (1.1) and the initial conditions (1.8) of the AL (1.3) satisfy: || φ (0) − ψ (0) || l ≤ C ε , (1.9) || φ (0) || l ≤ C γ, ε, (1.10) P µ (0) = X n ln(1 + µ | ψ n (0) | ) ≤ ln(1 + ( C µ, ε ) ) , (1.11) for some constants C , C γ, , C µ, > . Then, for arbitrary finite < T f < ∞ , there exists a constant C = C ( γ, µ, C , C µ, , C γ, , T f ) , such that the corresponding solutions for every t ∈ [0 , T f ] , satisfy the estimate || y ( t ) || l = || φ ( t ) − ψ ( t ) || l ≤ Cε . (1.12)For the proof of Theorem I.1 we use suitable estimates for the solutions of the AL lattice based on its deformedpower or norm , and energy arguments for the difference of solutions of the systems. It also makes essential use of theglobal existence of solutions for both lattices ensured by considering a physically relevant variant of a DNLS systemwhich combines both systems studied first in [38] (the so-called Salerno model or IN-DNLS, see [35], [38]). A secondvariant of Theorem I.1 is given in Theorem I.2.
When φ (0) = ψ (0) , under the assumptions (1.10) - (1.11) , for every ε > and t ∈ [0 , T f ] , the maximaldistance || y ( t ) || l ∞ = sup n ∈ Z | y n ( t ) | = sup n ∈ Z | ψ n ( t ) − φ n ( t ) | between individual units of the systems satisfies theestimate || y ( t ) || l ∞ ≤ ˜ Cε , (1.13) for some constant ˜ C = ˜ C ( γ, µ, C µ, , C γ, , T f ) . Theorem I.2 is proved with an alternative method which makes use of the Fourier transform of the local distancefunction of the solutions.Both results establish the closeness of the solutions to the AL and the DNLS as ε →
0, with explicit expressionsfor the associated constants C in dependence of the parameters of both systems; interestingly, we identify that thequantifications of the constants C and ˜ C derived from both approaches of the proofs are consistent (see remarkIV.1). The main application of both theorems is that they rigorously justify that at least, small amplitude localizedstructures provided by the analytical solutions of the integrable AL-lattice persist in the DNLS lattice. In other words,the DNLS lattice admits small amplitude solutions of the order O ( ε ), that stay O ( ε )-close to the analytical solutionsof the AL, for any 0 < T f < ∞ . In this regard, the analytical arguments show that the growth of the distance || y ( t ) || l is uniformly bounded for any ǫ > t ∈ (0 , ∞ ) as ddt || y ( t ) || l ≤ M ε , (1.14)( M depends on the parameters and initial data but not on t ), and consequently, the distance of solutions growths atmost linearly for any t ∈ (0 , ∞ ), as || y ( t ) || l ≤ M t ε . (1.15)In the case of Theorem I.2, where φ (0) = ψ (0), we have || y ( t ) || l ∞ ≤ ˜ M t ε , ∀ t ∈ (0 , ∞ ) . (1.16)In a similar context, we refer also to the linear time-growth estimates for the relevant distant function between thesolutions of the complex Ginzburg-Landau pde and the NLS pde, when the inviscid limit of the former is considered[36], which can even grow exponentially [37].The closeness results of Theorems I.1 and I.2 can be extended to other important cases of DNLS systems. Theseinclude the DNLS with generic power-nonlinearity F ( z ) = | z | σ z for σ > F ( z ) = z | z | . These extensions of Theorems I.1 and I.2 are given for the above models in higher-dimensional lattices Z N , N ≥
1. Note that for the AL and its generalizations in Z analytical localized solutions have been constructed[39],[40],[41],[42].To corroborate our analytical results we include a short numerical study treating the example of the soliton solution(1.4) when launched on DNLS lattice with the cubic nonlinearity (1.1), and the DNLS with saturable nonlinearity.The numerical findings are in excellent agreement with the theoretical results. Concerning the saturable nonlinearity,while analytically the magnitude of closeness is found to be weaker (of order O ( ε ) due to the fact the saturablenonlinearity is sublinear), the numerical results indicate an improved order though, similar to the cubic case.The presentation of the paper is as follows: Section II recalls some basic properties of the DNLS and the AL lattices,focusing on their conserved quantities and auxiliary results that will aid the main proofs. In Section III we provethe global existence result for the extended IN-DNLS of[35],[38]. Section IV contains the proofs of the main resultsTheorem I.1 and I.2, while section V deals with the extensions to N -dimensional lattices. In section VI we presentthe results of the numerical study. Section VII summarizes the findings and provide a brief plan for further relevantstudies to be still performed. II. PRELIMINARIES
For convenience, we set κ = ν = 1, without affecting the generality of the proofs, which are valid in either thefocusing or the defocusing case. The AL (1.3) can be derived from the Hamiltonian given by H = X n (cid:18) ψ n ( ψ n +1 + ψ n − ) − µ ln(1 + µ | ψ n | ) (cid:19) , (2.17)with the following deformed Poisson bracket (cid:8) ψ m , ψ n (cid:9) = (1 + µ | ψ m | ) δ m,n , { ψ m , ψ n } = (cid:8) ψ m , ψ n (cid:9) = 0 , (2.18)yielding the equation of motion as ˙ ψ n = { H, ψ } . (2.19)The following conserved (deformed) norm of the AL determined by P µ = X n ln(1 + µ | ψ n | ) , (2.20)is of particular importance for later studies.The DNLS can be derived from the Hamiltonian H = X n (cid:16) φ n ( φ n +1 + φ n − ) − γ | φ | (cid:17) , (2.21)using the standard Poisson bracket and the equation of motion i ˙ φ n = { H, φ } . (2.22)For the DNLS the norm P γ = X n ∈ Z | φ n | , (2.23)is conserved.The AL equation is completely integrable [6], whereas its DNLS counterpart (1.1) is known to be nonintegrable[4],[8]. Notice that in (1.3) and (1.1) the nonlinear terms are both of cubic order. However, they are markedly different in the sense that, the nonlinear terms in (1.3) are of nonlocal nature compared to the local terms in (1.1).In the case of the vanishing boundary conditions (1.5), the functional space setting is based on the spaces of complexsummable sequences l p = φ = ( φ n ) n ∈ Z ∈ C || φ || l p = X n ∈ Z | φ n | p ! /p . (2.24)For any φ = ( φ n ) n ∈ Z , ψ = ( ψ n ) n ∈ Z ∈ l we consider the inner product( φ, ψ ) l = X n ∈ Z φ n ψ n , (2.25)where ψ denotes the conjugate of ψ n . With the associated norm || φ || l = ( φ, φ ) , (2.26)( l , ( · , · ) , || · || ) is a complex Hilbert space. The following auxiliary result will aid the ensuing studies. Lemma II.1.
Assume that the initial condition (1.8) of the AL-lattice (1.3) is such that P µ (0) = X n ∈ Z ln(1 + µ | ψ n (0) | ) < ∞ . (2.27) Then, the corresponding solution of the AL lattice satisfies the estimate || ψ ( t ) || l = X n ∈ Z | ψ n ( t ) | ≤ exp( P µ (0)) − , ∀ t ≥ . (2.28) Proof:
Using that the function f : R + → R + , x ln(1 + x ) , (2.29)is continuous and bijective, we write P µ = X n ln(1 + µ | ψ n | ) = X n | λ n | . (2.30)From (2.30), we get the estimate: X n | ψ n | = X n (cid:0) exp( | λ n | ) − (cid:1) = X n ∞ X k =0 | λ n | k k ! − ! = X n ∞ X k =1 | λ n | k k ! = ∞ X k =1 k ! X n (cid:0) | λ n | (cid:1) k ≤ ∞ X k =1 k ! X n | λ n | ! k = ∞ X k =1 P kµ k != ∞ X k =0 P kµ k ! − P µ ) − . (2.31)Since P µ ( t ) is conserved, i.e P µ ( t ) = P µ (0) for all t ≥
0, it follows that X n ∈ Z | ψ n ( t ) | ≤ exp( P µ (0)) − , ∀ t ≥ , (2.32)and the proof is finished. (cid:3) III. GLOBAL EXISTENCE OF SOLUTIONS FOR THE IN-DNLS LATTICE
For the current study of existence and uniqueness of a global solution of the AL and DNLS, we combine them inthe Integrable-Nonintegrable-DNLS (IN-DNLS) single system, introduced first in [38]: i dψ n dt + (1 + µ | ψ n | )( ψ n +1 + ψ n − ) + γ | ψ n | ψ n = 0 , n ∈ Z , (3.1)with ψ n ∈ C and initial conditions: ψ n (0) = ψ n, , n ∈ Z . (3.2)Note that for γ = 0 ( µ = 0), the AL (DNLS) results from (3.1). The study of the IN-DNLS provided informationabout the intrinsic collapse of localized states in the presence of integrability-breaking terms, in particular, how thereflection symmetry and translational symmetry of the integrable AL are broken by the on-site nonlinearity of theDNLS. Thereby the study of the global existence of solutions of the IN-DNLS is an essential tool in the presentfunctional analytic set-up for the main closeness results of AL and DNLS.We start by noticing that for any ψ ∈ l the linear operator A : l → l ,( Aψ ) n ∈ Z = ψ n +1 + ψ n − , (3.3)is continuous, since || Aψ || l ≤ || ψ || l . (3.4)We formulate the infinite dimensional dynamical system (3.1)-(3.2) as as an initial value problem in the Hilbert space l (see [43]): ˙ ψ = F ( ψ ) ≡ i [(1 + µ | ψ | ) Aψ + γ | ψ | ψ ] , t > , (3.5) ψ (0) = ψ . (3.6)Regarding the global existence of a unique solution to (3.5)-(3.6), we have the following Proposition III.1.
For every ψ ∈ l , the problem (3.5)-(3.6) possesses a unique global solution φ ( t ) on [0 , ∞ ) belonging to C ([0 , ∞ ) , l ) . Proof:
First, we prove the local existence of a solution: For this aim, the system (3.1)-(3.2) is conveniently expressedas an equivalent system of integral equations ψ n ( t ) = ψ n (0) + i Z t (cid:2) (1 + µ | ψ n ( τ ) | )∆ ψ n ( τ )] + γ | ψ n ( τ ) | ψ n ( τ ) (cid:3) dτ, (3.7)with the notation ∆ ψ n = ψ n +1 + ψ n − . We consider the set B = (cid:8) φ ∈ C [0 , ˜ t ] , l ( Z ) | || φ || l ( Z ) ≤ κ (cid:9) , (3.8)which is a Banach space itself, with norm || φ || B = sup t ∈ [0 , ˜ t ] || φ || l ( Z ) . (3.9)Next, for φ ∈ l ( Z ), we define the nonlinear operator Q n ( φ ( t )) = φ n (0) + i Z t (cid:2) (1 + µ | φ n ( τ ) | )∆ φ n + γ | φ n ( τ ) | φ n ( τ ) (cid:3) dτ. (3.10)We shall prove that the operator Q establishes a contraction mapping on B . Note first, that it satisfies the upperbound || Q ( φ ) || B ≤ κ + ˜ t (cid:2) (1 + µκ )2 κ + γκ (cid:3) . (3.11)We may choose κ < κ/ t ≤ κ (1 + µκ )2 κ + γκ , (3.12)so that Q : B → B . Now, for every φ, ψ ∈ B , we have Q n ( φ ( t )) − Q n ( ψ ( t )) = i Z t (cid:0)(cid:2) (1 + µ | φ n ( τ ) | )∆ φ n ( τ ) + γ | φ n ( τ ) | φ n ( τ ) (cid:3) − (cid:2) (1 + µ | ψ n ( τ ) | )∆ ψ n ( τ ) + γ | ψ n ( τ ) | ψ n ( τ ) (cid:3)(cid:1) dτ = Z t (cid:0) (∆ φ n − ∆ ψ n ) + µ ( | φ n | ∆ φ n − | ψ n | ∆ ψ n ) − γ ( | φ n | φ n − | ψ n | ψ n ) (cid:1) dτ, (3.13)and estimate the norm as || Q ( φ ) − Q ( ψ ) || B ≤ ˜ t (cid:2) µ + 2 γ ) κ (cid:3) || φ − ψ || l . (3.14)Choosing ˜ t such that ˜ t ≤ min (cid:26) κ µκ ) κ + γκ ,
12 + (18 µ + 2 γ ) κ (cid:27) , (3.15)it is assured that Q is a contraction mapping on B . Then by Banach’s fixed point theorem, there exists a uniquesolution of (3.1) provided by the unique fixed point of Q . To justify that ψ ( t ) is C with respect to t , we see from(3.1) that sup t ∈ [0 , ˜ t ] || ˙ ψ ( t ) || l ≤ µκ ) κ + γκ . (3.16)Consequently, the solution belongs to C ([0 , ˜ t ] , l ). Then, constructing a maximal solution is achieved by repeatingthe procedure above with initial conditions ψ (˜ t − T f ) for some 0 < T f < ˜ t .To conclude with global existence of solutions, we remark that the Hamiltonian of (3.1) is H IN = X n ( ψ n ψ n +1 + ψ n ψ n +1 ) − γ X n | ψ n | − µ X n ln(1 + µ | ψ n | ) . (3.17)The deformed Poisson-brackets are { ψ n , ψ m } = i (1 + µ | ψ n | ) δ nm , { ψ n , ψ m } = { ψ n , ψ m } = 0 , and the equation of motion (3.1) is obtained as ˙ ψ n = { H IN , ψ n } . Then, global existence in l follows from the conservation of (3.17) and Lemma II.1. In particular, global existencefor the AL (1.3) is ensured by the conservation of P µ ( t ) = X n ln(1 + µ | ψ n ( t ) | ) , (3.18)so that the Lemma above entails that || ψ ( t ) || l < ∞ , ∀ t ≥
0. For the DNLS (1.1), global existence in l is establishedby the conservation of P γ ( t ) = X n | φ n ( t ) | . (3.19)This concludes the proof. (cid:3) IV. PROOFS OF CLOSENESS OF THE AL AND DNLS SOLUTIONS
Proof of Theorem I.1:
Closeness will be proved in the metric dist l ( ϕ, θ ) = || ϕ − θ || l , ∀ ϕ, θ ∈ l . We consider thelocal distance of the solutions y n = φ n − ψ n . On the one hand, we have that ddt || y ( t ) || l = 2 || y ( t ) || l ddt || y ( t ) || l , (4.1)while, on the other hand, we estimate the derivative of the l -norm as follows: ddt || y || l = X n (cid:26) i (cid:2) ( y n +1 + y n − ) y n − ( y n +1 + y n − ) y n (cid:3) + iµ | ψ n | (cid:2) ( ψ n +1 + ψ n − ) y n − ( ψ n +1 + ψ n − ) y n (cid:3) − iγ | φ n | ( φ n y n − φ n y n ) (cid:27) = 2 µ X n | ψ n | (cid:20) (Im ψ n +1 + Im ψ n − )Re y n − (Re ψ n +1 Re ψ n − )Im y n (cid:21) + 2 γ X n | φ n | (cid:20) Im y n Re φ n − Im y n Re φ n (cid:21) ≤ µ sup n | ψ n | X n ∈ Z (cid:20) | ψ n +1 | + | ψ n − | (cid:21) | y n | + 4 γ sup n ∈ Z | φ n | X n | φ n || y n |≤ (cid:0) µ || ψ ( t ) || l + 2 γ || φ ( t ) || l (cid:1) || y ( t ) || l . (4.2)For the estimate (4.2), we made use of the Cauchy-Schwarz and the continuous embeddings l r ⊂ l s , || φ || l s ≤ || φ || l r , ≤ r ≤ s ≤ ∞ . (4.3)Then, combining (4.1) and (4.2), one has for t > ddt || y ( t ) || l ≤ (cid:0) γ || φ ( t ) || l + 2 µ || ψ ( t ) || l (cid:1) . (4.4)Note that under the hypotheses (1.9)-(1.11), Proposition III.1 ensures that the right-hand side of (4.4) is uniformlybounded for all t ∈ [0 , ∞ ). Integrating the inequality (4.4) in the arbitrary interval [0 , T f ], and using the assumption(1.9) on the distance || y (0) || l = || φ (0) − y (0) || l of the initial data, we obtain that || y ( t ) || l ≤ (cid:0) γC ,γ + 2 µC ,µ (cid:1) T f ε + || y (0) || l ≤ (cid:0) γC ,γ + 2 µC ,µ (cid:1) T f ε + C ε . Hence, for the constant C = 2 (cid:0) γC ,γ + 2 µC ,µ (cid:1) T f + C , (4.5)we conclude with the claimed estimate (1.12). (cid:3) The proof of Theorem I.1 shows that the distance between the solutions of the AL and the DNLS measuredin terms of the l − metric remains small (bounded above by O ( ε )), compared to the l − norm of the solutionsthemselves. The proof of Theorem I.2 illustrates that analogous features for the l ∞ − norm (sup norm) determiningthe maximal distance between individual units. Alternatively to the energy argument implemented for the proof ofTheorem I.1, we shall use now an argument- which to our knowledge is novel in this context, based on the spatialFourier transform of the local distance of the solutions. Proof of Theorem I.2:
The time evolution of the local distance variable y n = φ n − ψ n is determinedby ˙ y n = ˙ φ n − ˙ ψ n = i (cid:2) (1 + µ | ψ | n )( ψ n +1 + ψ n − ) − ( φ n +1 + φ n − ) − γ | φ n | φ n (cid:3) = i (cid:2) y n +1 + y n − + µ | ψ | n ( ψ n +1 + ψ n − ) − γ | φ n | φ n (cid:3) ≡ i ( y n +1 + y n − ) + iF n . (4.6)Performing a spatial Fourier-transform in (4.6), y n ( t ) = 12 π Z π − π ˆ y q ( t ) exp( iqn ) dq, ˆ y q ( t ) = X n y n ( t ) exp( − iqn ) , (4.7)gives the system ˙ˆ y q ( t ) = 2 i cos( q )ˆ y q ( t ) + i ˆ F q ( t ) , (4.8)the formal solution of which, is given byˆ y q ( t ) = i Z t ˆ F q ( τ ) exp( − i cos( q )( t − τ )) dτ. (4.9)Applying the inverse Fourier-transform to (4.9) we obtain y n ( t ) = i π Z π − π Z t ˆ F q ( τ ) exp( − i cos( q )( t − τ )) dτ exp( iqn ) dq. (4.10)With the help of the Cauchy-Schwarz inequality and the continuous embeddings (4.3), we estimate d n ( t ) as follows: | y n ( t ) | = 12 π (cid:12)(cid:12)(cid:12)(cid:12)Z π − π Z t ˆ F q ( τ ) exp( − i cos( q )( t − τ )) dτ exp( iqn ) dq (cid:12)(cid:12)(cid:12)(cid:12) ≤ π Z π − π Z t (cid:12)(cid:12)(cid:12) ˆ F q ( τ ) (cid:12)(cid:12)(cid:12) dτ dq = 12 π Z π − π Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n F n ( τ ) exp( − iqn ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dτ dq ≤ π Z π − π Z t X n | F n ( τ ) | dτ dq = Z t X n | F n ( τ ) | dτ = Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X n µ | ψ n ( τ ) | ( ψ n +1 ( τ ) + ψ n − ( τ )) + γ | φ n ( τ ) | φ n ( τ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dτ ≤ µ Z t X n | ψ n ( τ ) | ! / X n | ψ n ( τ ) | ! / dτ + γ Z t X n | φ n ( τ ) | ! / X n | φ n ( τ ) | ! / dτ = 2 µ Z t || ψ ( τ ) || l || ψ ( τ ) || l dτ + γ Z t || φ ( τ ) || l || φ ( τ ) || l dτ ≤ µ Z t || ψ ( τ ) || l dτ + γ Z t || φ ( τ ) || l dτ ≤ (cid:16) µ (exp( P µ (0)) − / + γP γ (0) / (cid:17) t ≤ (cid:0) µC µ, + γC γ, (cid:1) t ε , ∀ t ∈ [0 , T f ] . (4.11)Taking the supremum over n ∈ Z one getssup n ∈ Z | y n ( t ) | ≤ (cid:0) µC µ, + γC γ, (cid:1) t ε ≤ (cid:0) µC µ, + γC γ, (cid:1) T f ε , ∀ t ∈ [0 , T f ] . (4.12)We set ˜ C = (2 µC µ, + γC γ, ) T f , (4.13)and the proof is completed. (cid:3) Remark IV.1.
We note the consistency of the constants C calculated in (4.5) and ˜ C given (4.13) : the latter isobtained when setting in C/ the value C = 0 , which is the case when φ (0) = ψ (0) . The fact that C > ˜ C is alsoconsistent with the embedding l ⊂ l ∞ . V. EXTENSIONS TO HIGHER DIMENSIONAL LATTICES AND OTHER NONLINEARITIES
In this section we provide extensions of the main results in higher dimensional lattices Z N , for N ≥ N -dimensional generalization of the AL.0 A. The case of DNLS with a generalised power nonlinearity.
The result of Theorem (I.1) can be extended to higher dimensional DNLS and AL lattices of the form i ˙ φ n + (∆ d φ ) n + γ | φ n | σ φ n = 0 , σ > , (5.1)and i ˙ ψ n + (∆ d ψ ) n + µ | ψ n | N X j =1 ( T j ψ ) n ∈ Z N = 0 , (5.2)respectively. The N -dimensional AL (5.2) is motivated by [44], where the case N = 2 is studied, as the specific limitof a 2D-generalization of the IN-DNLS 3.1. The operator (∆ d ψ ) n is the N -dimensional discrete Laplacian(∆ d ψ ) n ∈ Z N = X m ∈N n ψ m − N ψ n , where N n denotes the set of 2 N nearest neighbors of the point in Z N with label n . With the linear operator T j whichis defined for every ψ n , n = ( n , n , . . . , n N ) ∈ Z N , as( T j ψ ) n ∈ Z N = ψ ( n ,n ,...,n j +1 ,n j +1 ,...,n N ) + ψ ( n ,n ,...,n j − ,n j +1 ,...,n N ) , j = 1 , . . . , N, (5.3)the nonlocal nonlinearity in (5.2) generalizes the one of (1.3). When φ n ∈ C , the space l ( Z N ; C ) becomes a realHilbert space, l ( Z N ; C ) ≡ l ( Z N ; R ) × l ( Z N ; R ), if endowed with the real inner product( φ, ψ ) l = Re X n ∈ Z N φ n ¯ ψ n . In this setting the operator ∆ d : l → l is bounded and self-adjoint,(∆ d φ, ψ ) l = ( φ, ∆ d ψ ) l , φ, ψ ∈ l , (∆ d φ, φ ) l ≤ , || ∆ d φ || l ≤ N || φ || l . (5.4)It is also important to note that the embeddings (4.3) hold also in Z N . Theorem V.1.
A. Assume that σ > and that for every ǫ > , the initial conditions of the DNLS equation (5.1) and (5.2) satisfy || φ (0) − ψ (0) || l ≤ C ε , (5.5) || φ (0) || l ≤ C γ, ε, (5.6) P µ (0) = X n ∈ Z N ln(1 + µ | ψ n (0) | ) ≤ ln(1 + ( C µ, ε ) ) , (5.7) for some constants C , C γ, , C µ, > . Then, for arbitrary T f > , there exists a constant C = C ( γ, µ, C µ, , C γ, , T f , N ) , such that the corresponding solutions for every t ∈ [0 , T f ] , satisfy the estimate || y ( t ) || l = || φ ( t ) − ψ ( t ) || l ≤ C ε . (5.8)B. When φ (0) = ψ (0), under the assumptions (5.5)-(5.7), for every ε > t ∈ [0 , T f ], the maximal distance || y ( t ) || l ∞ = sup n ∈ Z N | y n ( t ) | = sup n ∈ Z N | ψ n ( t ) − φ n ( t ) | satisfies the estimate || y ( t ) || l ∞ ≤ Cε . (5.9) Proof : A. For global existence of solutions, we recall the following: The DNLS (5.1) conserves the l -norm (power) P γ = X n ∈ Z N | φ n ( t ) | , (5.10)1that is, || φ ( t ) || l = || φ (0) || l . (5.11)The generalization of the AL (5.2) conserves the deformed power P µ ( t ) = X n ∈ Z N ln(1 + µ | φ n ( t ) | ) , that is P µ ( t ) = P µ (0) . The higher dimensional version of Lemma II.1 is valid as well, implying under condition (5.7) the estimate || ψ ( t ) || l ≤ exp( P µ (0)) − ≤ C ,µ ε , ∀ t ≥ . (5.12)The above conservation laws together with the embedding (4.3), imply global existence of solutions of the systems(5.1) and (5.2), regardless of the dimension of the lattice and the nonlinearity exponent σ (see [43],[44] and remarkV.2 at the end of the section). Next, we consider again the local distance y n = φ n − ψ n , which satisfies the equation i ˙ y n = − ∆ d y n − γ | φ n | σ φ n − µ | ψ n | N X j =1 ( T j ψ ) n ∈ Z N . (5.13)When multiplying (5.13) by y n , summing over Z N , and keeping the imaginary parts, we then have that12 ddt || y || l = − γ Im X n ∈ Z N | φ n | σ φ n y n + µ Im X n ∈ Z N | ψ n | N X j =1 ( T j ψ ) n ∈ Z N y n . (5.14)For the first term on the right-hand side of (5.14), we have the estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im X n ∈ Z N | φ n | σ φ n y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) sup n ∈ Z N | φ n | (cid:19) σ X n ∈ Z N | φ n | | y n |≤ (cid:18) sup n ∈ Z N | φ n | (cid:19) σ || φ || l || y || l ≤ || φ || σ +1 l || y || l . (5.15)For the second term on the right-hand side of (5.14), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im X n ∈ Z N | ψ n | N X j =1 ( T j ψ ) n ∈ Z N y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ N sup n ∈ Z N | ψ n | X n ∈ Z N | ψ n | | y n |≤ N (cid:18) sup n ∈ Z N | ψ n | (cid:19) X n ∈ Z N | ψ n | | y n |≤ N (cid:18) sup n ∈ Z N | ψ n | (cid:19) || ψ || l || y || l ≤ N || ψ || l || y || l . (5.16)Then, from (5.14)-(5.16), we get the inequality ddt || y ( t ) || l ≤ (cid:0) γ || φ ( t ) || σ +1 l + 2 µN || ψ ( t ) || l (cid:1) , (5.17)which is the generalised, higher-dimensional analogue of (4.4) in the presence of the σ -nonlinearity of DNLS. With(5.17) in hand and the assumptions (5.5)-(5.7), we conclude the proof exactly as in the proof of Theorem I.1 with theconstant C modified as C = 2 (cid:0) γC ,γ + 2 µN C ,µ (cid:1) T f + C . B. The result is an immediate consequence of the embedding (4.3) and the estimate (5.8), since || y || l ∞ ≤ || y || l . (cid:3) B. DNLS with saturable nonlinearity.
Another important example concerns the DNLS with the saturable nonlinearityi ˙ φ n + (∆ d φ ) n + γφ n | φ n | = 0 . (5.18)For the saturable DNLS, numerous studies have verified the propagation of discrete solitons and the emergence ofbreathers in the 1D and 2D lattices [45],[46],[47],[48],[49]. The conserved quantities of the model are the power P γ ( t )(5.10) and the Hamiltonian H s = ( − ∆ d φ, φ ) l − γ X n ∈ Z N ln(1 + | φ n | ) . We establish closeness of the solutions of the saturable DNLS to those of the AL (5.2) according to
Theorem V.2.
A. Consider the DNLS equation (5.18) with the saturable nonlinearity. For every < ǫ < , weassume that the initial conditions of the AL (5.2) and the initial conditions of the DNLS (5.18) satisfy: || φ (0) − ψ (0) || l ≤ C ε, (5.19) || φ (0) || l ≤ C γ, ε, (5.20) P µ (0) = X n ln (cid:0) µ | ψ n (0) | (cid:1) ≤ ln (cid:0) C µ, ε ) (cid:1) , (5.21) for some constants C , C γ, , C µ, > . Then, for arbitrary T f > , there exists a constant C = C ( γ, µ, C µ, , C γ, , T f , N ) , such that the corresponding solutions for every t ∈ [0 , T f ] , satisfy the estimate || y ( t ) || l = || φ ( t ) − ψ ( t ) || l ≤ Cε. (5.22)
B. When φ (0) = ψ (0) , under the assumptions (5.20) - (5.21) , for every < ε < and t ∈ [0 , T f ] , we have the estimate || y ( t ) || l ∞ ≤ Cε. (5.23)
Proof:
The corresponding evolution equation for the local distance y n = φ n − ψ n , is i ˙ y n = − ∆ d y n − γφ n | φ n | − µ | ψ n | N X j =1 ( T j ψ ) n ∈ Z N , and the derivative of its l norm satisfies12 ddt || y || l = − γ Im X n ∈ Z N γφ n | φ n | y n + µ Im X n ∈ Z N | ψ n | N X j =1 ( T j ψ ) n ∈ Z N y n . (5.24)The first term on the right-hand side of (5.24) is estimated as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Im X n ∈ Z N γφ n | φ n | y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X n ∈ Z N | φ n | | y n | ≤ || φ || l || y || l , and the relevant differential inequality (5.17) is replaced by ddt || y ( t ) || l ≤ (cid:0) γ || φ ( t ) || l + 2 µN || ψ ( t ) || l (cid:1) . Using the fact that (5.11) holds also for the saturable model, we complete the proof as in Theorem I.1, underconsideration of the current assumptions. B. The proof is concluded as in the case of Theorem V.1 B . (cid:3) FIG. 1: Spatiotemporal evolution of the initial condition (6.28) in the DNLS lattice with γ = 1 for the cubic nonlinearity (leftpanel) and the saturable nonlinearity (right panel). Parameters of the initial condition a = π/
10 and β = 0 . µ = 1. Remark V.1.
As it is stressed above, the main application of the analytical results is that they establish that at leastsmall amplitude localized structures, as given by the (exact when N = 1 ) solutions of the (integrable when N = 1 ) AL,persist in the DNLS lattice. In more detail:1. The DNLS lattice with the power nonlinearity admits small amplitude solutions of the order O ( ε ) , that stay O ( ε ) -close to the solutions of the AL, for any ǫ > and any finite T f ∈ (0 , ∞ ) (see (1.14) - (1.16) ).2. The DNLS lattice with the saturable nonlinearity admits small amplitude solutions of the order O ( ε ) , that stay O ( ε ) -close to the solutions of the AL, for any < ǫ < and any finite T f ∈ (0 , ∞ ) . The latter restriction ofthe range of ǫ is due to the sub-linearity induced in the saturable case. Instead of (1.14) - (1.16) , we have for thedistance functions the following estimates: ddt || y ( t ) || l ≤ M ( γ, µ, C , C µ, , C γ, ) ε, (5.25) and consequently, the distance of solutions growths at most linearly for any t ∈ (0 , ∞ ) , as || y ( t ) || l ≤ M ( γ, µ, C , C µ, , C γ, ) tε. (5.26) For the distance || y ( t ) || l ∞ , we have || y ( t ) || l ∞ ≤ ˜ M ( γ, µ, C µ, , C γ, ) tε, ∀ t ∈ (0 , ∞ ) . (5.27) Remark V.2.
Crucial to our discussion of higher dimensional conservative discrete nonlinear Schr¨odinger equationsis the assurance of the global existence of their solutions. The solutions of the DNLS (5.1) exist unconditionally forany σ > and any N ≥ . This is a crucial difference to its NLS pde counterpart whose solutions may blow-up infinite time when σ > /N . The same unconditional global existence is shared by the higher dimensional AL (5.2) orIN-DNLS models. As explained in [44] for the 2D-lattice, if the value of the deformed power is P , then the sup-norm || φ || l ∞ of the solution can never exceed the value [exp( µP ) − . This is actually the argument of Lemma II.1, whichcan be extended to the case N ≥ , establishing global existence of solutions for the N -dimensional AL and IN-DNLSequations. The work [44] provides an analysis of the notion of quasi-collapse which can be observed in discrete systems:Blow-up in finite time for the the NLS pde corresponds to the effect of concentration of all energy in few sites in theconservative discrete NLS counterparts. The quasi-collapse supplies the mechanism for the emergence of very narrowself-trapped states in the lattice from its initial distribution, which however, do not exhibit a finite-time-singularity(unless additional energy gain mechanisms are present [50]). FIG. 2: Top row: Time evolution of || y ( t ) || l and || y ( t ) || l ∞ , corresponding to the soliton dynamics shown in the upper panelof Figure 1 for the DNLS with cubic and saturable nonlinearity (details in the text of section VI). Bottom row: Space timeevolution of the soliton center X CM = ( P n n | φ n | ) / ( P n | φ n | ) for the cubic and the saturable DNLS. VI. A NUMERICAL STUDY: PERSISTENCE OF THE AL-SOLITON IN DNLS
To illustrate the applicability of the analytical results we performed a numerical study examining the dynamics ofthe DNLS lattice for both types of nonlinearity, cubic and saturable. As initial conditions we used the one-solitonsolution of the AL (1.4): φ n (0) = ψ sn (0)= sinh β √ µ sech( βn ) exp( iαn ) , n ∈ Z , (6.28) || ψ s (0) || l = || φ (0) || l = ε, (6.29)where α ∈ [ − π, π ] and β ∈ [0 , ∞ ). In order to comply with the smallness condition (6.29), we chose the parametervalues accordingly so that persistence of the corresponding AL soliton in the DNLS can be expected. Figure 1 depictsthe spatio-temporal evolution of the density | φ n ( t ) | of the soliton initial condition when α = π/
10 and β = 0 . γ = 1. The dynamics for the cubic (saturated) DNLS is shown in the left (right) panel.The evolution is shown for the time span t ∈ [0 , T f = 2500) and for a chain of K = 2000 units with periodicboundary conditions. The evolution in both DNLS systems confirms the persistence of the AL soliton with amplitudeof order O ( ε ) in both lattices. Notably, persistence lasts for a significant large time interval, in particular with view5to that our analysis is a “continuous dependence on the initial data result” where generally, the time interval of sucha dependence on the initial data for a given equation might be short . Moreover, we studied the continuous dependenceof two different systems, the cubic and the saturable DNLS, respectively. The dynamics of the solitons are almostindistinguishable in both DNLS lattices. Note that for this example of initial condition, the value of the l -norm(6.29) of the initial condition is ε = 0 . y ( t ) = φ ( t ) − ψ ( t ) , of the solutions of the DNLS and the AL: Figure 2 depicts the time evolution of || y ( t ) || l (left panel) and || y ( t ) || l ∞ (right panel), corresponding to the dynamics of the cubic DNLS shown in the upper panel of Figure 1. The timeevolution for the cubic DNLS is plotted with the blue curve while for the saturable with red. However, the curves arestill indistinguishable, in conformity with the dynamics portrayed in Figure 1. Since for the considered example ofthe initial condition ǫ ∼ O (10 − ) and the time of integration is of order T f ∼ O (10 ), the constants C and ˜ C in theestimates (1.12) and (1.13), as calculated in (4.5) and (4.13), should be of the same order C ∼ O (1). Then, accordingto the analytical estimates one has for this large T f || y ( t ) || l ∞ ≤ || y ( t ) || l ≤ Cε ∼ O (1) , t ∈ [0 , T f ] , Figure 2 reveals that the numerical order is significantly lower: || y ( t ) || l ∼ O (10 − ), for t ∈ [0 , T f ]. The variation of || y ( t ) || l ∞ is even of smaller order as shown in the bottom panel, exhibiting that || y ( t ) || l ∞ ∼ O (10 − ). Variations ofthe same order, that is || y ( t ) || l ∼ O (10 − ), and || y ( t ) || l ∞ ∼ O (10 − ), are observed for the saturable DNLS; valueswhich lie even significantly lower than the analytically obtained upper bounds of Theorem V.2. Similarly, for thenumerical time growth-rate of || y ( t ) || l ddt || y ( t ) || l ∼ O (10 − ) , t ∈ [0 , T f ] , which is even smaller than the corresponding analytical prediction O (10 − ).Another interesting feature is the preservation of the soliton’s speed as shown in the bottom panel of Figure 2. Forthe considered value of β , we have sinh( β ) ≈ β , so the soliton’s speed is c = 2 sin( α ) = 0 . VII. CONCLUSIONS
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