The closeness of localized structures between the Ablowitz-Ladik lattice and discrete nonlinear Schrödinger equations: Generalized AL and DNLS systems

Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp The closeness of localized structures between the Ablowitz–Ladik lattice and discrete nonlinear Schrödinger equations: Generalized AL and DNLS systems Cite as: J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 Submitted: 22 September 2021 •Accepted: 30 March 2022 • Published Online: 19 April 2022 Dirk Hennig,1Nikos I. Karachalios,1,a) and Jesús Cuevas-Maraver2,3 AFFILIATIONS 1Department of Mathematics, University of Thessaly, Lamia GR 35100, Greece 2Grupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla, Escuela Politécnica Superior, C/ Virgen de África, 7, 41011 Sevilla, Spain 3Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain a)Author to whom correspondence should be addressed: [email protected] ABSTRACT The Ablowitz–Ladik system, being one of the few integrable nonlinear lattices, admits a wide class of analytical solutions, ranging from exact spatially localized solitons to rational solutions in the form of the spatiotemporally localized discrete Peregrine soliton. Proving a closeness result between the solutions of the Ablowitz–Ladik system and a wide class of Discrete Nonlinear Schrödinger systems in a sense of a continuous dependence on their initial data, we establish that such small amplitude waveforms may be supported in nonintegrable lattices for significantly large times. Nonintegrable systems exhibiting such behavior include a generalization of the Ablowitz–Ladik system with power-law nonlinearity and the discrete nonlinear Schrödinger equation with power-law and saturable nonlinearities. The outcome of numerical simulations illustrates, in excellent agreement with the analytical results, the persistence of small amplitude Ablowitz–Ladik analytical solutions in all the nonintegrable systems considered in this work, with the most striking example being that of the Peregine soliton. Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0072391 I. INTRODUCTION The Ablowitz–Ladik (AL) equation,1–3 i˙ ψn=κν(ψn+1−2ψn+ψn−1)+μ∣ψn∣2(ψn+1+ψn−1),n∈Z, (1.1) is the famous integrable discretization of the cubic nonlinear Schrödinger partial differential equation (PDE), i∂tu+νuxx +γ∣u∣2u=0, x∈R. (1.2) The integrability of (1.1) was proved by the discrete version of the inverse scattering transform.1The AL is one of the few known completely integrable (infinite) lattice systems admitting exact soliton solutions.4–7 The one-soliton solution of AL reads1,2 as follows: J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-1 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp ψs n=sinh β √μsech[β(n−ut)]exp(−i(ωt−αn)), ω=−2 cos αcosh β, u=2β−1sin αsinh β, (1.3) with α∈[−π,π]and β∈[0,∞). In Refs. 8and 9, it was shown that the AL equation (1.1) admits [again in similarity with the integrable Nonlinear Schrödinger (NLS) equation (1.2)] rational solutions as the discrete counterpart of the Peregrine soliton (AL-PS), ψr n(t)=q[1−4(1+q2)(1+4iq2t) 1+4n2q2+16q4t2(1+q2)]e2iq2t, (1.4) where the parameter qfixes a background amplitude. Such exact solutions do not exist for the the more relevant (with regard to applications in physics) non-integrable Discrete Nonlinear Schrödinger (DNLS) equation, i˙ ϕn+κν(ϕn+1−2ϕn+ϕn−1)+γ∣ϕn∣2ϕn=0, n∈Z,γ∈R. (1.5) However, in Ref. 10, we established the closeness of the solutions of the AL system (1.1) to the DNLS (1.5) in the l2and l∞-metrics in a sense of a continuous dependence on their initial data. The most significant application of this closeness result is that the DNLS lattice admits small amplitude solutions of the order O(ε)that stay O(ε3)-close to the analytical solutions of the AL for any finite time 0 <Tf<∞. Furthermore, taking the advantages offered by the discrete ambient space, we extended the result to higher dimensional lattices and other examples of nonlinearities. The numerical studies of Ref. 10, for the persistence of the AL-soliton in the cubic DNLS (1.5) and the saturable one (see below), corroborated to high accuracy the analytical predictions, that these non-integrable DNLS models sustain small amplitude solitary wavespreservingforsignificantlylargetimeintervals,thefunctionalformandcharacteristics(asthevelocity)oftheanalyticalAL-one-soliton solution. In the present paper, we provide further analytical and numerical extensions of the results of Ref. 10: On the analytical side, we establish the closeness of the solutions between the generalization of the Ablowitz–Ladik (G-AL) lattice, i˙ Un=κν(Un+1−2Un+Un−1)+δ∣Un∣2σ(Un+1+Un−1),n∈Z, (1.6) and the integrable AL (1.1). Our study in this paper for the G-AL (1.6) is motivated by the work in Ref. 11, where numerous features were explored, ranging from the derivation of conservation laws of the model, the numerical identification of discrete solitons, and the study of their stability—as the nonlinearity exponent σis varied—through extended Vakhitov–Kolokolov criteria, to the presentation of numerical evidence for quasi-collapse as indicated by the appearance of large amplitudes for large values of σ. The results of Ref. 11 justified that the G-AL (1.6) similar to the other important generalized DNLS models with power nonlinearity,12–14 F(z)z=γ∣z∣2σz, (1.7) is significant for the study of the transition from integrability when σ=1 to non-integrability for σ>1, and the potential structural changes to the dynamics of intrinsic localized modes possessed by these systems.13,15 In this regard, we think that our closeness approach is of particular relevance: it simultaneously establishes the existence and persistence of small amplitude localized waveforms for a general class of nonintegrable DNLS systems, which approximate the functional form of the analytical solutions of the integrable AL (1.1). As mentioned above, the analysis also covers the DNLS with saturable nonlinearity, F(z)z=γz 1+∣z∣2, (1.8) a model governing the dynamics of photo-refractive media; we refer to the works16–20 concerning the propagation and stability of discrete solitons. Thus, extra motivation is given to pursue further numerical studies exploring the closeness of solutions of the integrable AL to the solutions of the above generalized G-AL and DNLS systems. The presentation of this paper is as follows: in Sec. II, we recall useful information on the conserved quantities and the Hamiltonian structure of the AL and DNLS lattices studied herein. In Sec. III, we prove the closeness of the solutions between the integrable AL (1.1) and itsgeneralization(1.6).Theprooffollowsthe lines in Ref.10,aided this time, bythenontrivial conserved quantity oftheG-AL, which involves the Gauss hypergeometric function.11 We also prove that the solutions remain close in the sense of momenta. Section IV is devoted to the numerical studies and is divided into three parts. In Subsection IV A, we present the results of the numerical study concerning the persistence of small amplitude single-soliton solutions (1.3) of the AL in the quartic G-AL (1.6), corresponding to the case σ=2. It is shown that the AL-soliton persists when implemented as an initial condition to the G-AL for significantly large times. Furthermore, the numerical results for J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-2 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp the evolution of the norms of the distance function of the corresponding solutions not only fulfill but also turn out to be even sharper than the respective analytical estimates. A notable feature is that the evolving soliton in the G-AL lattice preserves with a high accuracy the speed of the analytical AL-soliton solution, as well as, its momentum. The same excellent agreement with the analytical predictions is illustrated in Subsection IV B for the dynamics of the analytical AL-soliton in the DNLS with the power nonlinearity (1.7) in its quintic case σ=2. We also test the limits of the analytical persistence results regarding the smallness of the amplitudes of the AL-soliton initial conditions in the cubic σ=1 and in the quintic DNLS σ=2. While the analytical estimates are again satisfied, we find that the persistence of the AL-soliton in these non-integrablelattices lastsfor significantlylower times(in comparisonto thesmaller amplitudecases); thisis inaccordancewiththeanalysis, as our closeness results rely, in the notion of continuous dependence, on the initial conditions. Furthermore, we observe the emergence of novel dynamical features: in the cubic DNLS, the AL-soliton evolves as a high time-period-spatially localized breather, while in the quintic DNLS, we observe a self-similar decay of the initial condition. Still, in both cases, we observe the preservation of the speed of the initial pulse. Subsection IV D concludes with the presentation of a particularly interesting application of the closeness of the solution approach: the persistenceofsmall amplituderationalsolutionsofthe formoftheAL-PS(1.4). Remarkably,andyetinexcellent agreementwiththeanalytical considerations, the numerical findings corroborate the existence/persistence of spatiotemporal localized waveforms significantly close to the AL-PS, in all the cases of the non-integrable lattices considered herein. Finally, Sec. Vsummarizes the main findings and gives an outlook to extensions for future studies. II. CONSERVED QUANTITIES AND SYMPLECTIC STRUCTURE OF THE AL AND DNLS SYSTEMS For simplicity, we set in (1.1), (1.5), and (1.6), κ=ν=1. It is useful to recall that all the lattice systems, which are subject of the present study, are particular cases of the general lattice dynamical system studied in Refs. 11,23, and 24, i˙ An+f(∣An∣2)(An+1+An−1)+F(∣An∣2)An=0. (2.1) Considering the functions g(ρ)=∫ρ 0 dy f(y),G(ρ)=∫ρ 0 F(y)dy f(y), it can be proved that the system admits a Hamiltonian structure with the Hamiltonian H=∑ n∈Z An(An+1+An−1)+∑ n∈N G(∣An∣2). Then, system (2.1) can be written as i˙ An=f(∣An∣2)∂H ∂An. System (2.1) has also the constant of motion P=∑ n∈Z g(∣An∣2), (2.2) known as the power (or the number, or norm). The Poisson bracket is given as {Φ,Ψ}=−i∑ n∈Z(∂Φ ∂An ∂Ψ ∂An−∂Φ ∂An ∂Ψ ∂An)f(∣An∣2), where Φand Ψare arbitrary functionals of the variable An. As a particular case, we have that {An,An}=−if(∣An∣2), and J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-3 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp {An,Am}={An,Am}=0. With respect to the Poisson bracket, system (2.1) is written in the following form: ˙ An={H,An}. Now, let us return to the particular cases of the DNLS systems studied herein. ●The AL equation (1.1): It corresponds to the case of system (2.1) for f(ρ)=1+μρ and F(ρ)=−2. We remark that the integrable AL conserves the momentum M=2Im∑ψnψn+1. (2.3) ●The generalization AL equation (1.6): It is the case of f(ρ)=1+μρσand F(ρ)=−2. ●The cubic DNLS equation (1.5): f(ρ)=1 and F(ρ)=−2+γρ. ●The DNLS equation with general power nonlinearity (1.7): f(ρ)=1 and F(ρ)=−2+γρσ. ●The DNLS equation with saturable nonlinearity (1.8): f(ρ)=1 and F(ρ)=−2+γ 1+ρ2. Another important case that we will not study in the present paper is the Salerno model (SM).25,26 It corresponds to the case of f(ρ) =1+μρ and F(ρ)=−2+γρ, that is, a combination of the AL and DNLS systems. The study of the Salerno model provides information about the intrinsic collapse of localized states in the presence of integrability-breaking terms, in particular, how the reflection symmetry and translational symmetry of the integrable AL are broken by the on-site nonlinearity of the DNLS. The review article24 highlights the relation between the AL and SM systems and establishes a link between the classical systems considered in the present paper, and their quantum counterparts in the form of Bose–Hubbard models. For the global in time existence of solutions of the SM, we refer to Ref. 10. For the proof of the main results, which are given in Sec. III, we will explicitly refer to the power (2.2) associated with the generalized G-AL system (1.6). It has a non-trivial functional form provided and studied in Ref. 11 and is a key ingredient for the implementation of the closeness arguments. III. PROOF OF CLOSENESS OF THE AL AND G-AL SOLUTIONS The analytical proof of closeness we shall demonstrate concerns the infinite lattice with vanishing boundary conditions. However, we remark that the argument can be easily modified to cover other cases of boundary conditions, such as periodic ones. Consider, then, the initial conditions for the integrable AL (1.1) and the non-integrable G-AL, ψn(0)=ψn,0,n∈Z, (3.1) and Un(0)=Un,0,n∈Z, (3.2) respectively. We start with the statement of the following theorem. Theorem III.1 a. We assume that for every 0<ε<1, the initial conditions (3.1) of the AL (1.1) and the initial conditions (3.2) of the G-AL (1.6) satisfy the following: ∥ψ(0)−U(0)∥l2≤C0ε3, (3.3) ∥U(0)∥l2≤Cδ,0 ε, (3.4) Pμ(0)=∑ nln(1+μ∣ψn(0)∣2)≤ln(1+(Cμ,0 ε)2), (3.5) for some constants C0,Cδ,0,Cμ,0 >0. Then, for arbitrary Tf>0, there exists a constant C =C(δ,μ,Cμ,0,Cδ,0,Tf,σ)such that the corresponding solutions for every t ∈[0,Tf],satisfy the estimate J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-4 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp ∥y(t)∥l2=∥ψ(t)−U(t)∥l2≤Cε3. (3.6) b. Under assumptions (3.4) and (3.5), for every 0<ε<1and t ∈[0,Tf], the maximal distance ∥y(t)∥l∞=supn∈Z∣yn(t)∣=supn∈Z∣ψn(t) −Un(t)∣between individual units of the systems satisfies the estimate ∥y(t)∥l∞≤Cε3. (3.7) Proof. For simplicity, we set κ=ν=1. The corresponding evolution equation for the local distance yn=ψn−Unreads i˙ yn=Δdyn+[μ∣ψn∣2(ψn+1+ψn−1)−δ∣Un∣2σ(Un+1+Un−1)], (3.8) where Δdyn=yn+1−2yn+yn−1is the 1D discrete Laplacian. Let us recall that when un∈C, the space l2(Z;C)becomes a real Hilbert space, l2(Z;C)≡l2(Z;R)×l2(ZN;R), if endowed with the real inner product (u,v)l2=Re ∑ n∈ZNun¯vn. In this setting, the operator Δd:l2→l2is bounded and self-adjoint, (Δdu,v)l2=(u,Δdv)l2,u,v∈l2, ∥Δdu∥2 l2≤4∥u∥2 l2. Moreover, it satisfies (Δdu,u)l2=−∑ n∈Z∣un+1−un∣2≤0 forall u∈l2. (3.9) When multiplying Eq. (3.8)byyn, summing over Z, and keeping the imaginary parts, we may use (3.10) and get that 1 2 d dt∥y(t)∥2 l2=μIm∑ n∈Z[∣ψn∣2(ψn+1+ψn−1)]yn−δIm∑ n∈Z[∣Un∣2σ(Un+1+Un−1)]yn. (3.10) To estimate the terms on the right-hand side of (3.10), we use the Cauchy–Schwarz inequality and the continuous embeddings lr⊂ls,∥ϕ∥ls≤∥ϕ∥lr, 1 ≤r≤s≤∞. (3.11) For the first term on the right-hand side of (3.10), we get ∣Im∑ n∈Z[∣ψn∣2(ψn+1+ψn−1)]yn∣≤2sup n∈Z∣ψn∣∑ n∈ZN∣ψn∣2∣yn∣ ≤2(sup n∈Z∣ψn∣)2∑ n∈ZN∣ψn∣∣yn∣≤2(sup n∈ZN∣ψn∣)2∥ϕ∥l2∥y∥l2 ≤2∥ψ∥3 l2∥y∥l2. (3.12) For the second term on the right-hand side of (3.10), we obtain ∣Im∑ n∈Z[∣Un∣2σ(Un+1+Un−1)]yn∣≤2sup n∈Z∣Un∣∑ n∈ZN∣Un∣2σ∣yn∣ ≤2(sup n∈Z∣Un∣)2σ∑ n∈ZN∣Un∣∣yn∣≤2(sup n∈ZN∣Un∣)2σ∥U∥l2∥y∥l2 ≤2∥U∥2σ+1 l2∥y∥l2. (3.13) Then, from (3.10)–(3.13), we get the inequality J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-5 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp 1 2 d dt∥y(t)∥2 l2≤2(μ∥ψ∥3 l2+δ∥U∥2σ+1 l2)∥y(t)∥l2. (3.14) Inequality (3.14), when combined with d dt ∥y(t)∥2 l2=2∥y(t)∥l2d dt ∥y(t)∥l2, implies that d dt∥y(t)∥l2≤2(γ∥ψ∥3 l2+δ∥U∥2σ+1 l2). (3.15) For the solutions of the AL, the deformed norm Pμ(t)=∑ nln(1+μ∣ψn∣2) is conserved, that is, Pμ(t)=Pμ(0). (3.16) This conservation implies the following bound (see Ref. 10, Lemma II.1 and Ref. 15): ∥ψ(t)∥2 l2≤exp(Pμ(0))−1≤C2 0,με2,∀t≥0. (3.17) On the other hand, the G-AL has a nontrivial conservation law,11 Pδ(t)=∑ n∈Z∣Un∣22F1(1, 1 σ,1 +1 σ;−∣Un∣2σ), (3.18) where 2F1(a,b,c;z)is the Gauss hypergeometric function for a,b,c,z∈C. We recall that 2F1(a,b,c;z)=1+ab cz+a(a+1)b(b+1) c(c+1)2! z2+⋅⋅⋅ (3.19) on the disk ∣z∣<1 and by analytic continuation outside this disk. By using the expansion (3.19) for the term 2F1(1, 1 σ,1 +1 σ;−∣Un∣2σ)of the quantity (3.18) and its conservation, as well as the embedding relation (3.11) for r=2 and s=2σ, we may derive the following inequality for small values of ∥U∥l2=(∑n∈Z∣U∣2)1 2: ∑ n∈Z∣Un(t)∣2+O(∑ n∈Z∣Un(t)∣2σ)=∑ n∈Z∣Un(0)∣2+O(∑ n∈Z∣Un(0)∣2σ) ≤∑ n∈Z∣Un(0)∣2+O((∑ n∈Z∣Un(0)∣2)σ) ≤C2 δ,0ε2+O(ε2σ), (3.20) for which we have also used the smallness assumption (3.4) on the initial data. Hence, under these assumptions, we deduce the estimate ∥U(t)∥2 l2≤ˆ C2 0,σ,δε2+O(ε2σ), (3.21) where ˆ C0,σ,δwill stand for a general constant depending on σand Cδ,0. Then, using (3.17) and (3.21) in the differential inequality (3.15), we get that d dt∥y(t)∥l2≤2(μC3 0,μ+δˆ C2σ+1 0,δ,σ)ε3. (3.22) Integrating the inequality (3.22) in the arbitrary interval [0,Tf]and using the assumption (3.3) on the distance ∥y(0)∥l2=∥ψ(0)−U(0)∥l2of the initial data, we obtain that J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-6 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp ∥y(t)∥l2≤2(μC3 0,μ+δˆ C2σ+1 0,σ,δ)Tfε3+∥y(0)∥l2 ≤2(μC3 0,μ+δˆ C2σ+1 0,σ,δ)Tfε3+C0ε3. Hence, for the constant C=2(μC3 0,μ+δˆ C2σ+1 0,σ,δ))Tf+C0, (3.23) we conclude with the claimed estimate (3.6). b. The result is an immediate consequence of embedding (3.11) and estimate (3.6) since ∥y(t)∥ł∞≤∥y(t)∥ł2.◻ Remark III.1 As in Ref. 10, we should also remark herein, that the Proof of Theorem III.1 establishes that the growth rate of the distance ∥y(t)∥l2is uniformly bounded for any 0<ϵ<1and t ∈(0,∞), that is, d dt∥y(t)∥l2≤Kε3, (3.24) where the constant K depends only on the parameters and the initial conditions, but not on t. Therefore, the distance of solutions grows at most linearly for any t ∈(0,∞)according to the estimate ∥y(t)∥l2≤Mtε3, (3.25) and similarly for the case of the l∞-distance, ∥y(t)∥l∞≤K1tε3,∀t∈(0,∞), (3.26) with the constant K1bearing a similar dependence as M. Such linear growth rates as (3.24) and (3.26) have been proved particularly relevant in the context of nonlinear PDEs. We refer to the time-growth estimates for the relevant distant function between the solutions, when the NLS is consdidered as the inviscid limit of the Ginzburg–Landau PDE.21 In such limits, the distance functions may grow even exponentially with respect to time.22 As the momentum M(2.3) is conserved only by the integrable AL system (1.1), having the closeness result of Theorem III.1 in hand, we may estimate the difference of momenta between the solutions of the integrable AL (1.1) and its non-integrable generalization (1.6). CorollaryIII.1. AssumethattheconditionsofTheoremIII.1aresatisfied.Then,forarbitraryTf>0,thedifferenceofmomentaΔM between the solutions of the integrable AL (1.1) and (1.6) satisfies for every T ∈[0,Tf], the estimate ΔM(t)≤Cε4(3.27) for some constant C >0depending on the same parameters and Tfas the constant C given in Theorem III.1. Proof. We denote by MAL(t)=2Im∑ n∈Z ψn(t)ψn+1(t) the momentum associated with the solutions of the AL lattice (1.1) and by MGAL(t)=2Im∑ n∈Z Un(t)Un+1(t) the momentum associated with the G-AL lattice (1.6). Then, the difference ΔM=MAL −MGAL can be written as ΔM=2Im∑ n∈Z ψn(ψn+1−Un+1)+2Im∑ n∈Z Un+1(ψn−Un). J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-7 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp Then, by using the Cauchy–Schwarz inequality and the estimates (3.17) for ψand (3.21) for Uand (3.6) for y=ψ−U, we get that ∣ΔM∣≤2∥ψ∥l2∥ψ−U∥l2+2∥U∥l2∥ψ−U∥l2 ≤2CC0,με4+2Cˆ C0,σ,δε4 ≤Cε4, and the proof is completed. ◻ IV. NUMERICAL RESULTS A. Persistence of the AL-soliton in the G-AL lattice In this section, we present a study of the generalized G-AL lattice (1.6) in the cubic case σ=2 similar to the one presented in Ref. 10 for the dynamics of the AL-soliton. As the initial condition, we use the one-soliton solution of the AL equation (1.3), Un(0)=ψs n(0) =sinh β √μsech(βn)exp(iαn),n∈Z, (4.1) ∥ψs(0)∥l2=∥U(0)∥l2=ε, (4.2) where α∈[−π,π]and β∈[0,∞). In order to comply with the smallness condition (4.2), we choose the parameter values accordingly so that persistence of the corresponding AL-soliton in the DNLS can be expected. Figure 1 depicts the spatiotemporal evolution of the density ∣Un(t)∣2of the soliton initial condition when μ=1, α=π/10, and β =arcsinh(0.02)for the G-AL equation (1.6) with δ=1. The evolution is shown for the time span t∈[0,2500](Tf=2500)and for a chain of K=2400 units with periodic boundary conditions. The evolution of the soliton in the G-AL depicted in the top-left panel of Fig. 1 confirms the persistence of the AL-soliton with amplitude of order O(ε). Notably, persistence lasts for a significant large time interval, particularly when one has in mind that our analysis is a “continuous dependence on the initial data result” where, generally, the time interval of such a dependence on the initial data for a given equation might be short. Note that for this example of initial condition, the value of its l2-norm (4.2) is ε=0.2. Our numerical results confirm convincingly the analytical predictions presented by Theorem III.1, concerning the distance y(t)=U(t)−ψ(t), between the solutions of the AL and the G-AL. The top-right panel of Fig. 1 depicts the time evolution of ∥y(t)∥l2, and the bottom-left panel depicts the evolution of ∥y(t)∥l∞, corresponding to the dynamics of the quintic G-AL shown in the top-left panel of Fig. 1. The time evolution of the norm of the distance function is plotted with a green solid curve. Both panels illustrate the qualitative and quantitative relevance of the growth estimates (3.24)–(3.26). Note that since U(0)=ψ(0), the constant Con the estimates (3.6) and (3.7) reads as C=2(μC3 0,μ+ˆ C0,σ,δ)Tf. Since the considered example of the initial condition ϵ∼O(10−1)and the time of integration is of order Tf∼O(103), the constant Cshould be of the order C∼O(1). Then, according to the analytical estimates, one has for this large Tf ∥y(t)∥l∞≤∥y(t)∥l2≤Cε3∼O(1),t∈[0,Tf]. The corresponding panels of Fig. 1 reveal that the numerical order is significantly lower: ∥y(t)∥l2∼O(10−1), for t∈[0,Tf]. The variation of ∥y(t)∥l∞is even of smaller order as shown in the bottom panel, showing that ∥y(t)∥l∞∼O(10−2). For the numerical time-growth rate of ∥y(t)∥l2, it holds that d dt∥y(t)∥l2∼O(10−4),t∈[0,Tf], which is yet of lower order than the corresponding analytical prediction O(10−3). J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-8 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp FIG. 1. Top-left panel: spatiotemporal evolution of the initial condition (4.1) in the G-AL lattice with δ=1 for the quintic non-linearity σ=2. Parameters of the initial condition α=π/10 and β=arcsinh(0.02),μ=1. Top-right panel and bottom-left panel: time evolution of ∥y(t)∥l2and ∥y(t)∥l∞, respectively, corresponding to the soliton dynamics showninthetop-leftpanel(detailsinthe text of Sec. IV). Bottom-right panel: space time evolution of the soliton center XCM =(∑nn∣ϕn∣2)/(∑n∣ϕn∣2)for the AL (1.1) (black dashed-dotted line) and G-AL (1.6) (green continuous line). The bottom-right panel of Fig. 1 depicts the preservation of the soliton’s velocity in the G-AL lattice as the corresponding curves for the G-ALevolution (green curve)and the analyticalsoliton (black curve)are indistinguishable. Fortheconsidered valueofβ, wehavesinh(β)≈β so that the soliton’s speed is c=2sin(α)=0.6181. B. The study of the evolution of the AL-soliton in the DNLS 1. The quintic DNLS The same excellent agreement with the analytical predictions for the persistence of the AL-soliton is observed in the case of the generalized DNLS with the power nonlinearity (1.7), i˙ ϕn+κν(ϕn+1−2ϕn+ϕn−1)+γ∣ϕn∣2σϕn=0, n∈Z,γ∈R,σ>0. (4.3) The analogous result to Theorem III.1 was proved in Ref. 10 even for higher-dimensional DNLS lattices, which is recalled herein for the sake of completeness. Theorem IV.1 A. Assume that σ>0and that for every ε>0, the initial conditions of the AL lattice (1.1) and DNLS equation (4.3) satisfy J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-9 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp V. CONCLUSIONS In this work, we have continued our analytical and numerical investigations of the closeness of solutions between the integrable Ablowitz–Ladik lattice and non-integrable lattices for which the discrete nonlinear Schrödinger equation serves as the basic underlying model. With our analytical studies, we have established the closeness of solutions, in the sense of a continuous dependence on their initial data, between the Ablowitz–Ladik system and its important generalization with an extended power-law nonlinearity. The numerical investigations have further explored the issue of closeness of solutions for the generalized Ablowitz–Ladik system and the discrete nonlinear Schrödinger equation with a power-law nonlinearity and with saturable nonlinearity. For all these examples of non-integrable systems, the numerical studies justified, in full accordance with analytical predictions, that small amplitude solitonic structures possessing the form of the analytical solutions of the Ablowitz–Ladik system can be excited in nonintegrable lattices and survive for significantly long time intervals. One of the most striking examples is the persistence of spatiotemporal waveforms close to the analytical discrete Peregrine soliton, in all of the above-mentioned non-integrable lattices. Future considerations may consider other DNLS systems important for physical applications (such as those relevant for the dynamics of granular crystals27), lattices incorporating gain and loss effects,28 and crucially, extensions of our closeness approach to second-order in time lattice dynamical systems (as an example, we mention the closeness between the integrable Toda lattice29 and Fermi–Pasta–Ulam–Tsingou lattices30,31 or discrete Klein–Gordon systems32). Such investigations are in progress, and the results will be reported elsewhere. ACKNOWLEDGMENTS Theauthorswouldlike to thank the referee for constructive comments and suggestions. J.C.-M.acknowledgessupportfromEU (FEDER program 2014-2020) through both Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (under Project Nos. P18-RT-3480 and US-1380977), and MICINN and AEI (under Project Nos. PID2019-110430GB-C21 and PID2020-112620GB-I00. AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1M. J. Ablowitz and J. F. Ladik, “Nonlinear differential-difference equations and Fourier analysis,” J. Math. Phys. 17, 1011–1018 (1976). 2M. J. Ablowitz and J. F. Ladik, “A nonlinear difference scheme and inverse scattering,” Stud. Appl. Math. 55, 213–229 (1976). 3B. M. Herbst and M. J. Ablowitz, “Numerically induced chaos in the nonlinear Schrödinger equation,” Phys. Rev. Lett. 62, 2065 (1989). 4M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, New York, 1991). 5L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons (Springer-Verlag, Berlin, 1987). 6V. E. Vekslerchik, “Functional representation of the Ablowitz–Ladik hierarchy,” J. Phys. A: Math. Gen. 31, 1087–1099 (1998). 7K.-i. Maruno and Y. Ohta, “Casorati determinant form of dark soliton solutions of the discrete nonlinear Schrödinger equation,” J. Phys. Soc. Jpn. 75, 054002 (2006). 8A. Ankiewicz, N. Akhmediev, and J. M. Soto-Crespo, “Discrete rogue waves of the Ablowitz-Ladik and Hirota equations,” Phys. Rev. E 82, 026602 (2010). 9N. Akhmediev and A. Ankiewicz, “Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation,” Phys. Rev. E 83, 046603 (2011). 10D. Hennig, N. I. Karachalios, and J. Cuevas-Maraver, “The closeness of the Ablowitz-Ladik lattice to the discrete nonlinear Schrödinger equation,” J. Differ. Equations 316, 346–363 (2022). 11J.Cuevas-Maraver, P.G.Kevrekidis, B.A. Malomed,andL. Guo,“Solitary waves inthe Ablowitz–Ladikequationwith power-lawnonlinearity,” J.Phys.A: Math.Theor. 52, 065202 (2019). 12E. W. Laedke, K. H. Spatschek, and S. K. Turitsyn, “Stability of discrete solitons and quasicollapse to inrinsically lcalized modes,” Phys. Rev. Lett. 73, 1055–1059 (1994). 13O.Bang,J.J.Rasmussen,andP. L. Christiansen,“SubcriticallocalizationinthediscretenonlinearSchrödingerequationwitharbitrarypowernonlinearity,”Nonlinearity 7, 205–218 (1994). 14P. Pacciani, V. V. Konotop, and G. Perla Menzala, “On localised solutions of discrete nonlinear Schrödinger equation: An exact result,” Physica D 204, 122–133 (2005). 15P. L. Christiansen, Yu. B. Gaididei, V. K. Mezentsev, S. L. Musher, K. Ø. Rasmussen, J. J. Rasmussen, I. V. Ryzhenkova, and S. K. Turitsyn, “Discrete localized states and localization dynamics in discrete nonlinear Schrödinger equations,” Phys. Scr. 1996(T67), 160–166. 16J. Cuevas and J. C. Eilbeck, “Discrete soliton collisions in a waveguide array with saturable nonlinearity,” Phys. Lett. A 358, 15–20 (2006). 17L. Hadzievski, A. Maluckov, M. Stepi´ c, and D. Kip, “Power controlled solitons stability and steering in lattices with saturable nonlinearity,” Phys. Rev. Lett. 93, 033901 (2004). 18M. Stepic, D. Kip, L. Hadzievski, and A. Maluckov, “One-dimensional bright discrete solitons in media with saturable nonlinearity,” Phys. Rev. E 69, 066618 (2004). 19R. A. Vicencio and MM. Johansson, “Discrete soliton mobility in two dimensional waveguide arrays with saturable nonlinearity,” Phys. Rev. E 73, 046602 (2006). 20T. R. O. Melvin, A. R. Champneys, P. G. Kevrekidis, and J. Cuevas, “Radiationless travelling waves in saturable nonlinear Schrödinger lattices,” Phys. Rev. Lett. 97, 124101 (2006). J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-16 Published under an exclusive license by AIP Publishing Journal of Mathematical Physics ARTICLE scitation.org/journal/jmp 21J. Wu, “The inviscid limit of the complex Ginzburg–Landau equation,” J. Differ. Equations 142, 413–433 (1998). 22T. Ogawa and T. Yokota, “Uniqueness and inviscid limits of solutions for the complex Ginzburg-Landau equation in a two-dimensional domain,” Commun. Math. Phys. 245, 105–121 (2004). 23M. H. Hays, C. D. Levermore, and P. D. Miller, “Macroscopic lattice dynamics,” Physica D 79, 1–15 (1994). 24O. Dutta, M. Gajda, P. Hauke, M. Lewenstein, D.-S. Lühmann, B. A. Malomed, T. Sowi´ nski, and J. Zakrzewski, Non-standard Hubbard models in optical lattices: A review, Rep. Prog. Phys. 78, 066001 (2015). 25M. Salerno, “Quantum deformations of the discrete nonlinear Schrödinger equation,” Phys. Rev. A 46, 6856–6859 (1992). 26D. Cai, A. R. Bishop, and N. Grønbech-Jensen, “Localized states in discrete nonlinear Schrödinger equation,” Phys. Rev. Lett. 72, 591–595 (1994). 27G. James, “Travelling breathers and solitary waves in strongly nonlinear lattices,” Philos. Trans. R. Soc. London, Ser. A 376, 20170138 (2018). 28Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83, 247–305 (2011). 29M. Toda, Theory of Nonlinear Lattices (Springer-Verlag, Berlin, Heidelberg, 1989). 30Y. V. Lvov and M. Onorato, “Double scaling in the relaxation time in the β-Fermi-Pasta-Ulam-Tsingou model,” Phys. Rev. Lett. 120, 144301 (2018). 31G. P. Berman and F. M. Izrailev, “The Fermi–Pasta–Ulam problem: Fifty years of progress,” Chaos 15, 015104 (2005). 32S. Flach and A. V. Gorbach, “Discrete breathers—Advances in theory and applications,” Phys. Rep. 467, 1–116 (2008). J. Math. Phys. 63, 042701 (2022); doi: 10.1063/5.0072391 63, 042701-17 Published under an exclusive license by AIP Publishing