On the proximity of Ablowitz–Ladik and discrete nonlinear Schrödinger models: A theoretical and numerical study of Kuznetsov-Ma solutions

Wave Motion 137 (2025) 103547 Available online 30 March 2025 0165-2125/© 2025 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Wave Motion journal homepage: www.elsevier.com/locate/wamot On the proximity of Ablowitz–Ladik and discrete nonlinear Schrödinger models: A theoretical and numerical study of Kuznetsov-Ma solutions Madison L. Lytle a,b, Efstathios G. Charalampidis a,c,∗, Dionyssios Mantzavinos d, Jesus Cuevas-Maraver e,f, Panayotis G. Kevrekidis g, Nikos I. Karachalios h aMathematics Department, California Polytechnic State University, San Luis Obispo, 93407-0403, CA, USA bDepartment of Applied Mathematics and Statistics, Colorado School of Mines, Golden, 80401, CO, USA cDepartment of Mathematics and Statistics, and Computational Science Research Center, San Diego State University, San Diego, 92182-7720, CA, USA dDepartment of Mathematics, University of Kansas, Lawrence, 66045, KS, USA eGrupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla. Escuela Politécnica Superior, C/ Virgen de África, 7, 41011 Sevilla, Spain fInstituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain gDepartment of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, 01003-4515, MA, USA hDepartment of Mathematics, University of Thessaly, Lamia, 35100, Greece A R T I C L E I N F O Keywords: Breathers Rogue waves Integrable and non-integrable lattice systems Bifurcation analysis A B S T R A C T In this work, we investigate the formation of time-periodic solutions with a non-zero background that emulate rogue waves, known as Kuznetsov-Ma (KM) breathers, in physically relevant lattice nonlinear dynamical systems. Starting from the completely integrable Ablowitz–Ladik (AL) model, we demonstrate that the evolution of KM initial data is proximal to that of the non-integrable discrete Nonlinear Schrödinger (DNLS) equation for certain parameter values of the background amplitude and breather frequency. This finding prompts us to investigate the distance (in certain norms) between the evolved solutions of both models, for which we rigorously derive and numerically confirm an upper bound. Finally, our studies are complemented by a two-parameter (background amplitude and frequency) bifurcation analysis of numerically exact, KM-type breather solutions of the DNLS equation. Alongside the stability analysis of these waveforms reported herein, this work additionally showcases potential parameter regimes where such waveforms with a flat background may emerge in the DNLS setting. 1. Introduction and motivation The study of dispersive nonlinear lattice dynamical models has been a topic of considerable interest over the past decades [1,2]. Among the physical areas motivating the relevant developments, one can single out the particular contributions from the study of optical waveguides [3] (but also continuum photorefractive media with periodic potentials) and the exploration of mean-field atomic Bose–Einstein condensates (BECs) in the presence of periodic external (so-called optical lattice) potentials [4]. However, ∗Corresponding author at: Department of Mathematics and Statistics, and Computational Science Research Center, San Diego State University, San Diego, 92182-7720, CA, USA. E-mail addresses: [email protected] (M.L. Lytle), [email protected] (E.G. Charalampidis), [email protected] (D. Mantzavinos), [email protected] (J. Cuevas-Maraver), [email protected] (P.G. Kevrekidis), [email protected] (N.I. Karachalios). https://doi.org/10.1016/j.wavemoti.2025.103547 Received 20 December 2024; Received in revised form 12 March 2025; Accepted 20 March 2025 Wave Motion 137 (2025) 103547 2 M.L. Lytle et al. relevant models, computations, and experiments are by no means limited to these subfields; rather, they broadly extend to other contexts, including, among others, nonlinear variants of electrical circuits [5], elastically interacting beads within granular crystal metamaterials [6–8], superconducting Josephson junction arrays [9,10], micromechanical arrays of cantilevers [11], and DNA denaturation models [12]. There has been a considerable wealth of models relevant to these spatially discrete applications, including Klein–Gordon and Fermi–Pasta–Ulam–Tsingou [13,14]. However, the most universal dispersive lattice nonlinear model is, arguably, the discrete nonlinear Schrödinger (DNLS) equation [15,16]. It serves as the prototypical vehicle for the study of solitary waves, instabilities, and dynamics in discrete nonlinear optics [17] while also holding considerable relevance in atomic BECs [4]. To this day, it continues to play a substantial role in modern developments concerning, e.g., topological lattices [17], flat bands [18,19], and many others. From a mathematical physics perspective, it has an additional, particularly interesting feature: the existence of an integrable analog [20]. This, in turn, creates the potential for numerous studies concerning the breaking of integrability, perturbation effects on conservation laws, solitary features, etc. in suitable interpolations between the integrable and non-integrable variants of the model [21]. An aspect of DNLS solitary waves that has been of interest in recent years is its potential to feature large-amplitude rogue or freak waves. The study of such waves [22] has recently attracted considerable attention due to the emergence of experimental capabilities that enable the detection and visualization of these patterns at the level of nonlinear waves in optical systems [23–26], fluid settings in water tanks [27–29], plasmas [30], and even BECs [31]. These developments have been summarized in a wide range of reviews, including [32–36]. In the discrete realm, the work of [37] showcased the existence of all the central nonlinear waveforms in the integrable Ablowitz–Ladik (AL) model: i.e., the Peregrine soliton (P) [38], the Akhmediev breather (AB) [39], as well as the Kuznetsov-Ma (KM) breather [40,41]. The emergence of such patterns in discrete integrable media raised the question of their potential observability [42,43]. When considering models interpolating between the integrable (AL) and non-integrable (DNLS) limit, these works set the expectation that extreme events are more likely to exist in the former, rather than in the latter. On the other hand, in recent years, some of the present authors have explored the proximity between the DNLS and the AL model through analytical estimates [44,45] as well as continuations between the two via numerical computations [46]. The present work aims to bring these different elements of the literature together, operating at the nexus of extreme solutions of lattice nonlinear dynamical systems of both the integrable (AL) and the non-integrable (DNLS) kind, and exploring the potential continuation of KM solutions from the former towards the latter. Moreover, the present work considers such questions from the complementary perspectives of rigorous estimates, as well as of numerical computations involving the existence, spectral stability, and nonlinear dynamics of such states. We find that the relevant states can often be continued and we can identify corresponding waveforms in the DNLS limit. This is corroborated by the presence of the analytical estimates. It is envisioned that the findings herein may motivate optical waveguide experiments that may enable the identification of the KM solutions presented herein. Such experiments should be directly accessible in arrays of optical waveguides [3], a topic of continuing interest in recent investigations [47]. They may also arise in an effective form in Bose–Einstein condensates in optical lattices, as explained, e.g., in [48]. Of particular interest in that regard will be the evolution of the relevant states and, more specifically, the manifestation of their dynamical instability, in connection with the results presented below. Our presentation is structured as follows. In Section 2, we present the model setup and relevant solutions of interest. In Section 3, we analyze some preliminary numerical computations showcasing the proximity over long times (for suitable solution parameters) of the KM waveforms for the AL and the DNLS models. This finding then motivates Section 4, where we provide a rigorous analysis of the growth over time of the distance between the solutions of the DNLS and the AL systems in the case of non-zero boundary conditions. We establish that this growth is linear in time with a prefactor whose dependence on the size of initial data is analyzed. In Section 5 we present a continuation of the KM branch of solutions of the DNLS model, along with a discussion of the spectral stability of the pertinent solutions. The work is summarized and our conclusions are presented in Section 6. 2. The model setup In our subsequent theoretical and computational analysis, we consider the Salerno model [21], explicitly given by i 𝛹𝑛+𝐶(𝛥𝑑𝛹)𝑛+𝑔(𝛹𝑛+1 +𝛹𝑛−1)|𝛹𝑛|2+ 2 (1 − 𝑔)|𝛹𝑛|2𝛹𝑛= 0,(1) that interpolates between the completely integrable Ablowitz–Ladik (AL) model [49,50] (at 𝑔= 1) and the discrete Nonlinear Schrödinger (DNLS) equation [15] (at 𝑔= 0). In Eq. (1), 𝛹𝑛=𝛹𝑛(𝑡) represents the complex-valued wavefunction at a lattice site 𝑛∈Z and time 𝑡∈R, (𝛥𝑑𝛹)𝑛=𝛹𝑛+1 − 2𝛹𝑛+𝛹𝑛−1, and the coupling constant 𝐶 between nearest neighboring sites is given by 𝐶= 1∕ℎ2 with ℎ being the lattice spacing, i.e., the distance between adjacent sites. Our main interest is the study of breather solutions of Eq. (1) on a non-zero background. Since such solutions involve two frequencies, i.e., one associated with the breather itself and the other with background oscillations, we consider the separation of variables ansatz: 𝛹𝑛=𝜓𝑛𝑒2𝑖𝑞2𝑡, 𝑞 > 0, 𝜓𝑛∈C,(2) where 𝑞 represents the background amplitude of the solution of interest, and thus 2𝑞2 is the oscillation frequency of the background. Upon inserting Eq. (2) to Eq. (1), we obtain: i𝜓𝑛+𝐶(𝛥𝑑𝜓)𝑛+𝑔(𝜓𝑛+1 +𝜓𝑛−1)|𝜓𝑛|2+ 2 [(1 − 𝑔)|𝜓𝑛|2−𝑞2]𝜓𝑛= 0,(3) Wave Motion 137 (2025) 103547 3 M.L. Lytle et al. which is the principal model equation of interest for studying time-periodic solutions 𝜓𝑛(𝑡) = 𝜓𝑛(𝑡+𝑇𝑏) of period 𝑇𝑏= 2𝜋∕𝜔𝑏 and frequency 𝜔𝑏. For 𝐶= 1, the AL model (𝑔= 1) admits explicit rogue wave solutions [37] including the (discrete) Kuznetsov-Ma (KM) breather, Peregrine (P) soliton, and (spatially-periodic) Akhmediev breathers (AB). In particular, the KM breather solution to the AL model is a time-periodic solution with period 𝑇𝑏(= 2𝜋∕𝜔𝑏), and is explicitly given by: 𝜓𝑛(𝑡) = 𝑞cos (𝜔𝑏𝑡+ i𝜃)+𝐺cosh (𝑟𝑛) cos (𝜔𝑏𝑡)+𝐺cosh (𝑟𝑛),(4) with parameters 𝜃= − arcsinh (𝜔𝑏∕(2𝑞2)), 𝑚= (1 + 𝑞2)∕𝑞2, 𝑟= arccosh [(cosh 𝜃+𝑚− 1)∕𝑚], and 𝐺= −𝜔𝑏∕(2𝑞2√𝑚sinh 𝑟). Note that the Peregrine breather [37,38] is the limiting case 𝑇𝑏↦∞ (or 𝜔𝑏↦0) of the KM solution of Eq. (4). In this work, we solely focus on studies revolving around KM-type solutions to both the AL (𝑔= 1) and DNLS (𝑔= 0) models. Hereafter, we fix 𝐶= 1 corresponding to a lattice with unit spacing, i.e., ℎ= 1. 3. Preliminary numerical studies Recently, in [46], the homotopic continuation of KM breathers (and their stability) in the Salerno model was considered, and numerical results showcased the emergence of KM-type breathers. Some of these KM-type solutions could be continued all the way to the DNLS limit (i.e., 𝑔= 0 in Eq. (1)), but featured an oscillatory background. Attempts for identifying such solutions on a flat background by starting from the anti-continuum limit (i.e., 𝐶= 0) are reported in [51], although the relevant waveforms obtained do not enjoy rogue-like behavior. That is to say, the waveforms therein do not seem to ‘‘appear out of nowhere and disappear without a trace’’ [52]. Herein, we take a different path and examine the possibility of identifying KM solutions to the DNLS by considering the recent advances on the closeness of localized structures between the AL and DNLS models [44,45,53]. In [44], the proximity of Peregrine waveforms in the AL and DNLS models was examined, showing their persistence in DNLS models when 𝑞, i.e., the value of the background, is small. Motivated by this finding, we now explore numerically whether KM breathers with small background amplitudes 𝑞 ≪ 1 and frequencies 𝜔𝑏≪1, i.e., close to the Peregrine limit, may persist in the DNLS model. In our computations that follow in this section, we utilize a lattice [−𝑁∕2, 𝑁∕2 − 1] consisting of 𝑁= 600 nodes (unless stated otherwise), where periodic boundary conditions are supplemented in Eq. (3). Then, the initial-value problem (IVP) consisting of Eqs. (3)–(4) is solved numerically by using MATLAB’s built-in Adams– Bashforth–Moulton ode113 solver (with relative and absolute tolerances 10−13). We checked our numerical results by additionally using the Bulirsch–Stoer method [54] (with same relative and absolute tolerances), and the results matched exactly. The fidelity of our computations utilizing both numerical methods was further checked by monitoring the conserved quantities for the AL and DNLS models, respectively given by 𝑃AL(𝑡) = ∑ 𝑛 log (1 + |𝜓𝑛|2),(5) and 𝑃DNLS(𝑡) = ∑ 𝑛|𝜓𝑛|2.(6) We found that the maximum relative errors, i.e., max (|𝑃AL(𝑡) − 𝑃AL(0)|∕𝑃AL(0)) (and similarly for the DNLS) were ∼ 10−14. Let us now turn our focus to the spatio-temporal evolution of KM breathers [cf. Eq. (4)] in Eq. (3) (again with 𝐶= 1) for 𝑔= 1 (AL) and 𝑔= 0 (DNLS). We use the analytical KM solution of Eq. (4) with 𝜔𝑏= 0.005 and 𝑞= 0.005 as an initial condition for both models at 𝑡= 0, and advance Eq. (3) forward in time for up to 20 periods, i.e., 𝑡= 20 × 𝑇𝑏≈ 2.5133 × 104. Our results for these cases are summarized in Fig. 1. In particular, the panel (a) of the figure depicts the temporal evolution of the amplitude at the 𝑛= 0 site, i.e., |𝜓0(𝑡)|, for the AL and DNLS models (i.e., 𝑔= 1 and 𝑔= 0 in Eq. (3)). It can be discerned from this panel that the evolutions of the amplitudes are close for a few periods of the time integration although a phase lag in their evolution gradually gets manifested, and progressively becomes more and more pronounced for longer times. We complement this first round of numerical simulations in panels (b) and (c) of Fig. 1 where we summarize the spatio-temporal evolution of the amplitude |𝜓𝑛(𝑡)| of the solutions for the AL and DNLS models, respectively. Despite this phase lag reported in panel (a), panels (b) and (c) exhibit similar qualitative (and quantitative) features in the evolution of the waveforms. These findings now beg the question whether such a KM solution can be identified numerically as a fixed point, i.e., as a numerically exact periodic orbit for the DNLS case. To explore this possibility, we followed the numerical approach of [46,51] in which a 𝑇𝑏= 2𝜋∕𝜔𝑏 time-periodic solution is sought by considering the Fourier decomposition 𝜓𝑛= ∞ ∑ 𝑚=−∞ 𝑛,𝑚𝑒i𝑚𝜔𝑏𝑡,(7) where 𝑛,𝑚 are the Fourier coefficients and 𝑚 are the Fourier modes in time. Then, upon plugging Eq. (7) into Eq. (3) we obtain a root-finding problem for 𝑛,𝑚 (see [46] for the explicit form of the relevant root-finding problem), that is solved with high accuracy by means of Newton’s method. We employ stopping criteria in the nonlinear residual and successive iterates of 10−14. It should be noted that the infinite sum of Eq. (7) is truncated according to |𝑚|≤𝑘 where 𝑘= 41, and thus 2𝑘+ 1 = 83 Fourier modes were employed. Wave Motion 137 (2025) 103547 4 M.L. Lytle et al. Fig. 1. (Color online) The spatio-temporal evolution of KM breathers in the AL and DNLS models [cf. Eq. (3) for 𝑔= 1 and 𝑔= 0, respectively] with 𝑞=𝜔𝑏= 0.005 in Eq. (4). Panel (a) compares the evolution of the amplitude at the 𝑛= 0 site, i.e., |𝜓0(𝑡)| with solid red (AL) and blue (DNLS) lines (see, the legend therein). Panels (b) and (c) showcase the spatio-temporal evolution of the amplitude |𝜓𝑛(𝑡)| of the KM solution in the AL and DNLS models, respectively. A lattice of 𝑁= 600 sites was used with periodic boundary conditions. Fig. 2. (Color online) Summary of results for the numerically exact, KM solution to the DNLS with 𝑞=𝜔𝑏= 0.005. The converged KM profile is presented in panel (a) where the spatial distribution of its amplitude is shown. The respective Floquet stability analysis results are presented in panel (b) where a real yet very weakly unstable mode exists with magnitude ≈ 1.0000086 (data not shown). The spatio-temporal evolution of the profile of panel (a) is shown in panel (c) showcasing its robustness over 20 periods. Finally, the maximum difference of the amplitudes at each time instant between the exact KM solution of Eq. (4) and numerically evolved KM of the DNLS equation is shown in panel (d). Upon using the exact KM breather of Eq. (4) as an initial guess for Newton’s method, our nonlinear solver converged to the profile shown in Fig. 2(a). We performed a Floquet stability analysis of this solution following the setup of the variational equations for the associated monodromy matrix as discussed in [46]. The respective results are shown in Fig. 2(b) where the red dots correspond to the Floquet multipliers 𝜆=𝜆𝑟+i𝜆𝑖. We find that all the multipliers lie on the unit circle (depicted with a solid black line) except for a real eigenvalue with a tiny positive real part (data not shown) associated with a real instability. Per the given lattice of 𝑁= 600 Wave Motion 137 (2025) 103547 5 M.L. Lytle et al. Fig. 3. (Color online) MI analysis of the CW solution to the DNLS for 𝑞=𝜔= 0.005. In particular, real unstable Floquet multipliers, i.e., 𝜆𝑟>1 (and 𝜆𝑖= 0) are probed as a function of the number of lattice sites 𝑁. The first unstable mode of CW solutions appears when 𝑁 > 628, and more and more unstable modes emerge with increasing 𝑁. sites and temporal discretization with 83 Fourier modes, this instability appears to be rather weak as its magnitude is ≈ 1.0000086. We check this finding in Fig. 2(c) which showcases the spatio-temporal evolution of the amplitude |𝜓𝑛(𝑡)| of the numerically exact KM breather of Fig. 2(a) over 20 periods. The solution itself is quite robust, maintaining its general characteristics over the time interval of integration of the DNLS we considered. We perform a comparison between the evolution of the exact KM breather of Eq. (4) and the numerically obtained one for the DNLS model in Fig. 2(d). In particular, we monitor the maximum difference (in its absolute value) of the amplitude of the exact KM solution and numerically exact KM one to the DNLS at each time instant. It can be discerned from the panel that despite being time-periodic solutions to different models themselves, i.e., AL and DNLS, they are quite proximal to one another (notice the order of the infinity norm of the difference, i.e., ∼ 10−5). Finally, we would like to comment on an important observation of our spectral stability analysis results. It is known from [55], that plane wave solutions (CW) to the focusing DNLS model are modulationally unstable (a stability calculation of CW solutions to the Salerno model that contains the AL is presented in [46] too). The KM breathers (alongside with P and AB ones) are mounted atop a finite yet modulationally unstable background (per the focusing DNLS and AL) where localized solutions may perturb the CW Floquet spectrum. In light of the spectral stability analysis results of Fig. 2(b) and the discretization used in the present calculation, remarkably, no unstable modes are observed other than the weakly unstable one mentioned previously. We also note that the time-translation mode 1 + 0i is accurately resolved in our computations. We investigated the absence of instabilities of the KM breather of Fig. 2(a) by performing a modulational instability (MI) analysis of the CW with 𝑞=𝜔𝑏= 0.005 parametrically as a function of the number of sites 𝑁. Our respective results are shown in Fig. 3 where we probe the real unstable modes 𝜆𝑟 as a function of 𝑁. In conjuction with the spectrum of Fig. 2(b) with 𝑁= 600, the MI analysis for the same 𝑁, surprisingly does not predict the presence of CW unstable modes. In fact, this is true for all 𝑁 ≲ 628. However, for a value of 𝑁 just above 𝑁= 628, we observe the emergence of the first unstable mode, and in general, the number of unstable modes starts getting increased with 𝑁 [cf. Fig. 3], i.e., we obtain a band of unstable modes as we approach the infinite lattice. This finding in turn suggests that for solutions sitting atop a finite yet small background, the computation of their Floquet spectrum requires the use of a very large number of sites for resolving it well. Despite the absence of a band of real unstable modes in the Floquet spectrum of the KM breather of Fig. 2(b) and in the spectra presented next, this observation indicates that the KM solutions that we identify in this work are expected to be MI unstable as 𝑁 increases. Having presented our preliminary numerical results to motivate the notion of proximity between the AL and DNLS models for small enough data, we analytically explore this proximity in the next section. 4. Theoretical justification of the proximal KM dynamics between AL and DNLS 4.1. The case of the infinite lattice For our subsequent theoretical analysis, we will introduce notation to distinguish the AL and DNLS limits of the Salerno model of Eq. (1). Recall that the AL model corresponds to the 𝑔= 1 case: i 𝛹𝑛+𝐶(𝛥𝑑𝛹)𝑛+(𝛹𝑛+1 +𝛹𝑛−1)|𝛹𝑛|2= 0,(8) whereas the DNLS model to the 𝑔= 0 case: i 𝛷𝑛+𝐶(𝛥𝑑𝛷)𝑛+ 2|𝛷𝑛|2𝛷𝑛= 0,(9) that is, 𝛹 and 𝛷 will stand for the solutions to the AL and DNLS models, respectively. Motivated by [53], the systems will be supplemented with general non-zero boundary conditions of the form lim |𝑛|→∞𝛹𝑛(𝑡) = lim |𝑛|→∞𝑒2𝑖𝑞2𝑡𝑞𝑛,lim |𝑛|→∞𝛷𝑛(𝑡) = lim |𝑛|→∞𝑒2𝑖𝑞2𝑡𝑞𝑛, 𝑡 ≥0,(10) Wave Motion 137 (2025) 103547 6 M.L. Lytle et al. where 𝑞𝑛={𝑞−, 𝑛 ≤0, 𝑞+, 𝑛 > 0,(11) and 𝑞± are complex constants with |𝑞±|=𝑞. The boundary conditions of Eqs. (10)–(11) can be transformed to be time independent via the change of variables 𝛹𝑛(𝑡) = 𝑒2𝑖𝑞2𝑡𝜓𝑛(𝑡), 𝛷𝑛(𝑡) = 𝑒2𝑖𝑞2𝑡𝜙𝑛(𝑡).(12) Using the change of variables in Eq. (12), we can rewrite Eqs. (8) and (9) in the form 𝑖 𝜓𝑛+𝐶(𝛥𝑑𝜓)𝑛− 2𝑞2𝜓𝑛+|𝜓𝑛|2(𝜓𝑛+1 +𝜓𝑛−1)= 0,(13) 𝑖 𝜙𝑛+𝐶(𝛥𝑑𝜙)𝑛+ 2 [|𝜙𝑛|2−𝑞2]𝜙𝑛= 0.(14) Note that the initial conditions remain unchanged under the change of variables of Eq. (12), namely, 𝜓𝑛(0) = 𝛹𝑛(0) and 𝜙𝑛(0) = 𝛷𝑛(0). The boundary conditions become lim 𝑛→±∞ 𝜓𝑛(𝑡) = lim 𝑛→±∞ 𝜙𝑛(𝑡) = 𝑞±,(15) so that lim|𝑛|→∞|𝜓𝑛(𝑡)|= lim|𝑛|→∞|𝛷𝑛(𝑡)|=𝑞 > 0. Next, we perform a second change of variables, 𝜓𝑛(𝑡) = 𝑈𝑛(𝑡) + 𝑞𝑛, 𝜙𝑛(𝑡) = 𝑉𝑛(𝑡) + 𝑞𝑛,(16) and Eqs. (13)–(14) become 𝑖 𝑈𝑛+𝐶(𝛥𝑑𝑈)𝑛+𝐶(𝛥𝑑𝑞)𝑛− 2𝑞2(𝑈𝑛+𝑞𝑛) +|𝑈𝑛+𝑞𝑛|2(𝑈𝑛+1 +𝑈𝑛−1 +𝑞𝑛+1 +𝑞𝑛−1)= 0,(17) 𝑖 𝑉𝑛+𝐶(𝛥𝑑𝑉)𝑛+𝐶(𝛥𝑑𝑞)𝑛+ 2 [|𝑉𝑛+𝑞𝑛|2−𝑞2](𝑉𝑛+𝑞𝑛) = 0.(18) The initial conditions for Eqs. (17)–(18), become 𝑈𝑛(0) = 𝜓𝑛(0) − 𝑞𝑛, 𝑉𝑛(0) = 𝜙𝑛(0) − 𝑞𝑛,(19) and the boundary conditions are zero at infinity, i.e., lim 𝑛→±∞ 𝑈𝑛(𝑡) = lim 𝑛→±∞ 𝑉𝑛(𝑡) = 0, 𝑡 ≥0.(20) Along the lines of [56] (see also [53] for the continuous counterpart), we can prove that, for any initial conditions 𝑈(0), 𝑉 (0) ∈ 𝓁2, there exist 𝑇∗ 𝐴𝐿(𝑈(0)), 𝑇 ∗ 𝐷𝑁𝐿𝑆 (𝑉(0)) >0 such that the above Cauchy problems for the modified AL and DNLS Eqs. (17) and (18) have unique solutions 𝑈∈𝐶1([0, 𝑇 ∗ 𝐴𝐿(𝑈(0))],𝓁2) and 𝑉∈𝐶1([0, 𝑇 ∗ 𝐷𝑁𝐿𝑆 (𝑉(0))],𝓁2). Starting from this local existence result and following the methods of [44,45,53], we can further prove that these solutions stay close to one another in the following sense: Theorem 4.1. Consider the Cauchy problems for the modified AL and DNLS models given by Eqs. (17) and (18) when supplemented with the initial conditions (19) and the vanishing boundary conditions at infinity (20). Let any 0< 𝜀 < 1 and assume that the initial conditions 𝑈(0) and 𝑉(0) have 𝓁2-norms of (𝜀), the 𝓁2-distance ‖𝑈(0) − 𝑉(0)‖𝓁2 between them is of (𝜀3), and that the background amplitude 𝑞 is of (𝜀), namely there exist constants 𝐶𝑖, 𝑖 = 1,…,4 ‖𝑈(0)‖𝓁2≤𝐶1𝜀, ‖𝑉(0)‖𝓁2≤𝐶2𝜀, ‖𝑈(0) − 𝑉(0)‖𝓁2≤𝐶3𝜀3,and 𝑞≤𝐶4𝜀. (21) Then, there exists 𝑇𝑐>0 and a positive constant 𝐶 > 0 such that the 𝓁2-distance between the solutions satisfies simultaneously the upper bounds ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2≤𝐶𝜀3𝑡, ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2≤ 𝐶𝜀, for all 0< 𝑡 ≤𝑇𝑐.(22) Theorem 4.1 is proved in Appendix. The time 𝑇𝑐 in the above proximity result is obtained as follows. Since 𝑈∈𝐶1([0, 𝑇 ∗ 𝐴𝐿],𝓁2), i.e., it is continuously differentiable with respect to time, the assumption (21) implies that there exists a time  𝑇𝐴𝐿 ∈ (0, 𝑇 ∗ 𝐴𝐿] such that the size of 𝑈(𝑡) is also of (𝜀), that is ‖𝑈(𝑡)‖𝓁2≤𝐶5𝜀, for all 𝑡∈ [0, 𝑇𝐴𝐿],(23) for some constant 𝐶5>0. Similarly, there is time  𝑇𝐷𝑁𝐿𝑆 ∈ (0, 𝑇 ∗ 𝐷𝑁𝐿𝑆 ] such that ‖𝑉(𝑡)‖𝓁2≤𝐶6𝜀, for all 𝑡∈ [0, 𝑇𝐷𝑁𝐿𝑆 ],(24) for some constant 𝐶6>0. Then, the proximity time 𝑇𝑐 is defined as 𝑇𝑐= min{ 𝑇𝐴𝐿, 𝑇𝐷𝑁𝐿𝑆 }. Wave Motion 137 (2025) 103547 7 M.L. Lytle et al. 4.2. The case of periodic boundary conditions As described in Section 3, in the numerical simulations we supplement the lattice (3) with periodic boundary conditions. This scenario bears some significant differences and is more tractable than the case of the infinite lattice discussed above, due to the conservation laws of the systems. The phase spaces for the periodic lattice are the spaces of periodic sequences with period 𝑁, denoted by 𝓁𝑝 𝑝𝑒𝑟 ∶= {𝑈= (𝑈𝑛)𝑛∈Z∈R∶𝑈𝑛=𝑈𝑛+𝑁,‖𝑈‖𝓁𝑝 per ∶= (ℎ 𝑁−1 ∑ 𝑛=0 |𝑈𝑛|𝑝)1 𝑝<∞},1≤𝑝≤∞, where ℎ denotes the lattice spacing. For simplicity, we set ℎ= 1 (as in the numerical study which corresponds to the choice 𝐶=1 ℎ2= 1 made above). We begin with the following analog of Theorem 4.1, which can be proved in exactly the same way as the results given in [44,45]. Theorem 4.2. Consider the Cauchy problems for Eqs. (13) and (14) when supplemented with periodic boundary conditions. Let any 0<𝜀<1 and assume that the initial conditions 𝜓(0) and 𝜙(0) have 𝓁2 𝑝𝑒𝑟-norms of (𝜀), the 𝓁2 𝑝𝑒𝑟-distance ‖𝜓(0) − 𝜙(0)‖𝓁2 𝑝𝑒𝑟 between them is of (𝜀3), i.e. there exist positive constants 𝐶𝑖, 𝑖= 1,…,4 such that ‖𝜓(0)‖𝓁2 𝑝𝑒𝑟 ≤𝐶1𝜀, ‖𝜙(0)‖𝓁2 𝑝𝑒𝑟 ≤𝐶2𝜀, ‖𝜓(0) − 𝜙(0)‖𝓁2 𝑝𝑒𝑟 ≤𝐶3𝜀3.(25) Then, for arbitrary 0< 𝑇 < ∞, there exists a constant  𝐶 > 0 such that the 𝓁2 𝑝𝑒𝑟-distance between the solutions satisfies the estimate ‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2 𝑝𝑒𝑟 ≤ 𝐶𝜀3𝑡, for all 0< 𝑡 ≤𝑇 . (26) The main difference between Theorems 4.1 and 4.2 is that in the latter case the conservation laws ensure the global existence of solutions, thus implying the validity of the proximity result (26) for arbitrary times. In fact, thanks to global existence, we also have the following corollary to Theorem 4.1, which will be used to justify theoretically the results of the numerical simulations shown in Fig. 5 for the case of the non-vanishing boundary conditions at infinity: Corollary 4.1. Consider the modified AL and DNLS models (17) and (18) supplemented with the initial conditions (19) and periodic boundary conditions. Let any 0< 𝜀 < 1 and assume that the initial conditions 𝑈(0) and 𝑉(0) have 𝓁2 per -norms of (𝜀), the 𝓁2 per -distance ‖𝑈(0) − 𝑉(0)‖𝓁2 per between them is of (𝜀3), and that the background amplitude 𝑞 is of (𝜀), namely there exist constants 𝐶𝑖, 𝑖 = 1,…,4 ‖𝑈(0)‖𝓁2 per ≤𝐶1𝜀, ‖𝑉(0)‖𝓁2 per ≤𝐶2𝜀, ‖𝑈(0) − 𝑉(0)‖𝓁2 per ≤𝐶3𝜀3,and 𝑞≤𝐶4𝜀. (27) Then, for arbitrary finite 0< 𝑇 < ∞, there exists positive constant 𝐶 > 0 such that the 𝓁2 per -distance between the solutions satisfies simultaneously the upper bounds sup 𝑡∈[0,𝑇 ]‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2 per ≤𝐶𝜀3.(28) The proof of Corollary 4.1 is identical to the one of Theorem 4.1 given in Appendix, the only difference being that the norm used is that of 𝓁2 per instead of 𝓁2. 4.3. Connection to the numerical findings of Section 3 Theorem 4.1 (for the infinite lattice with nonzero boundary conditions) and Theorem 4.2 (for the finite lattice with periodic boundary conditions) justify theoretically that the distance between the solutions of the AL and DNLS equations grows linearly at a rate which is at most of (𝜀3) when the initial and background data are of (𝜀). In the case of Theorem 4.1, according to the well-posedness results of [53,57], initial data and background of (𝜀) guarantee that the 𝓁2-norm of the solutions remains of (𝜀) over a lifespan of (𝜀−2) ≃ 𝑇𝑐. However, beyond that lifespan, the solutions may not remain of (𝜀) and, therefore, there may exist a time 𝑇∗ 𝑐 such that the distance of solutions may escape from the trapezoidal region delimited by the two upper bounds in (22). This scenario is portrayed by the left diagram of Fig. 4. On the other hand, in the case of Theorem 4.2 for the periodic problem, the (𝜀3) growth rate of the difference of solutions is transient. Indeed, recalling the conserved quantities (5) and (6), it was shown in [44] that ‖𝜓(𝑡)‖2 𝓁2 𝑝𝑒𝑟 ≤exp(𝑃AL(0)) − 1 ≲ 𝜀2,for all 𝑡≥0,(29) and, furthermore, due to the conservation of 𝑃DNLS, 𝑃DNLS(𝑡) = ‖𝜙(𝑡)‖2 𝓁2 𝑝𝑒𝑟 =‖𝜙(0)‖2 𝓁2 𝑝𝑒𝑟 =𝑃DNLS(0) ≲ 𝜀2,for all 𝑡≥0.(30) Hence, by the triangle inequality, ‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2 𝑝𝑒𝑟 ≤‖𝜓(𝑡)‖𝓁2 𝑝𝑒𝑟 +‖𝜙(𝑡)‖𝓁2 𝑝𝑒𝑟 ≤ 𝐶𝜀, for all 𝑡≥0,(31) Wave Motion 137 (2025) 103547 8 M.L. Lytle et al. Fig. 4. (Color online) The behavior of the distance according to the theoretical upper bounds of Theorems 4.1–4.2. Left: The case of nonzero boundary conditions. Here, the distance of solutions may increase beyond (𝜀) at finite time. Right: The case of periodic boundary conditions. Here, according to the uniform bound of inequality (31), the distance of solutions is at most (𝜀) for all times. It should be noted that, in certain scenarios, the distance curve (in blue) may be much closer to the 𝑡-axis than what is depicted above, at least for an initial time interval — e.g. see the corresponding graphs of Fig. 5. where the constant  𝐶 is independent of 𝑡. Consequently, the linear growth upper bound in the proximity estimate (26) is relevant only for times such that  𝐶𝜀3𝑡≤ 𝐶𝜀, i.e. 𝑡≤ 𝐶  𝐶 1 𝜀2=∶ 𝑇𝑢𝑏.(32) That is, 𝑇𝑢𝑏 defines the upper bound for the times where the linear growth of the distance between the solutions holds. For 𝑡 > 𝑇𝑢𝑏, the distance between the solutions should satisfy the uniform upper bound (31). This situation is illustrated in the right diagram of Fig. 4. Under the (𝜀) smallness conditions on the initial and background data (21) and (25), Theorem 4.1 for the infinite lattice supplemented with the nonzero boundary conditions (10)–(11) and Theorem 4.2 for the periodic lattice justify that the DNLS lattice admits solutions of (𝜀) with the following properties: 1. They diverge at most linearly in time from the analytical solutions of the AL lattice in terms of the 𝓁2-metric, with a linear growth rate at most of (𝜀3), for finite times in [0, 𝑇𝑐] in the case of the infinite lattice (left panel of Fig. 4) or satisfying at most the upper bound 𝑇𝑢𝑏 given by (32) in the periodic case (right panel of Fig. 4) . 2. For all 𝑡≥0, the solutions remain close to solutions of the AL lattice– at most within (𝜀) with respect to the 𝓁2-metric – for all 𝑡∈ [0, 𝑇𝑐] in the case of the infinite lattice and for all 𝑡 > 0 in the periodic case. 3. For the distance measured in all 𝓁𝑝-metrics with 𝑝≥2, due to the embedding 𝓁𝑞⊂𝓁𝑝,‖𝑢‖𝓁𝑝≤‖𝑢‖𝓁𝑞,1≤𝑞≤𝑝≤∞,(33) it is reasonable to expect an even smaller rate of proximity than the one for the 𝓁2-metric for the times described above. For example, when 𝑝= ∞, which is relevant for a comparison of the amplitudes of solutions, it is theoretically justified to expect an improved rate of proximity. In summary, comparing the results of Theorems 4.1 and 4.2, in the case of the infinite lattice there is a possibility that, at some finite time 𝑇∗ 𝑐≥𝑇𝑐≃(𝜀−2), the distance of solutions will escape from the trapezoidal region capped by the (𝜀) dashed horizontal line (see Fig. 4). This is due to the fact that, in the case of nonzero boundary conditions, it is not known whether the solutions are uniformly bounded. On the other hand, in the case of periodic boundary conditions, the solutions individually remain bounded by the initial data of size (𝜀) due to the conservation of 𝑃DNLS and 𝑃AL; therefore, by the triangle inequality, the distance of solutions is guaranteed to remain within the trapezoidal region below the (𝜀) dashed horizontal line at all times (see Fig. 4). However, since we are interested in applying these analytical results for the KM solutions (4), it is crucial to underline the following remarks regarding the behavior of the distance of solutions in 𝓁2 per : •It is evident by the change of variables (16) that ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2 per =‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2 per .(34) Due to Corollary 4.1, the distance ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2 per approximates well the norm ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2 of the solutions 𝑈(𝑡) and 𝑉(𝑡) of the modified equations Eqs. (17) and (18) when supplemented with the initial conditions (19) and the vanishing boundary conditions at infinity (20). So we expect in the numerical simulations, that ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2 per will satisfy the upper bounds (22) of Theorem 4.1 for all 0< 𝑡 < 𝑇𝑐, under the smallness conditions on the initial data (27). •Due to Eq. (34), the distance ‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2 per will also satisfy the upper bounds (22) of Theorem 4.1 for all 0<𝑡<𝑇𝑐. In addition, due to the middle estimate of (29), the conservation law (30) and the middle estimate of (31), will satisfy for all Wave Motion 137 (2025) 103547 9 M.L. Lytle et al. Fig. 5. Summary of numerical results for the AL and DNLS models [cf. Eqs. (13) and (14)] in conjuction with our theoretical analysis on their proximity. We evolve KM initial data with (𝑞, 𝜔𝑏, 𝜀) ≃ (0.09,0.01,0.054) and (𝑞, 𝜔𝑏, 𝜀) ≃ (0.34,0.46,0.425) in the top and bottom rows, respectively, over two periods. Panels (a) and (d), and (b) and (e) depict the spatio-temporal evolution of the amplitudes |𝜓𝑛| and |𝜙𝑛|, respectively. The panels (c) and (f) summarize the comparisons made between the theoretical estimates on the proximity of the models with the numerically obtained distance (see, the legend and insets therein) of the solutions after subtracting the background 𝑞 (i.e. corresponding to the modified Eqs. (17) and (18)). 𝑡 > 0, the uniform in time bound ‖𝜓(𝑡) − 𝜙(𝑡)‖𝓁2 𝑝𝑒𝑟 ≤√exp(𝑃AL(0)) − 1 + √𝑃DNLS(0) ∶= ℎper .(35) For solutions satisfying non-zero boundary conditions as the KM (4), ℎper can be quite large (given the exponentiation involved in the relevant expression). In order to have ℎper =(𝜀) as in the last inequality of (31), we must assume the smallness conditions of (𝜀) for 𝑃AL(0) and 𝑃DNLS(0) to achieve the final smallness estimates of (𝜀2) given in (29) and (30). Thus, the dashed horizontal line in the right panel of Fig. 4 which is 𝑦=ℎper becomes quantitatively useful for solutions on a non-vanishing background 𝑞 as the KM, only when 𝑞 is sufficiently small, and is of (𝜀) only for 𝑞=(𝜀) with 𝜀 ≪ 1. Therefore, for the KM solutions, the scenario depicted in the left panel of Fig. 4 is the one which is quantitatively useful to be explored for small and moderate values of 𝑞 < 1. It is remarkable that the numerical findings of Section 3 for the case of the KM solutions showcase that the growth rate of the distance is significantly lower than the ones associated with the theoretical bounds, and the order of proximity is much higher than what is predicted theoretically. This is in accordance with several numerical findings for both discrete [44,45] and continuous [53] setups, which illustrate that the rate of divergence and proximity may depend on the particular analytical solution of the integrable system considered. In many cases (bright solitons [44], fast soliton collisions [53]), this rate is of (𝜀𝑝) with 𝑝 > 3. Indicatively, and in line with the numerical study of Fig. 1, the rate of divergence of the solutions was found to be in the neighborhood of 𝜀5. We conclude our theoretical analysis by showcasing two examples comparing the theoretical estimates with the numerical computations in Fig. 5. Note that, for computational purposes, we may consider the assumptions (21) with constants 𝐶𝑖= 1, 𝑖= 1,…,4. Then, we may consider the relevant times  𝑇𝐴𝐿, 𝑇𝐷𝑁𝐿𝑆 such that the bounds (23) and (24) hold with the constants 𝐶5=𝐶6= 1. For this choice of unit constants 𝐶𝑖, we may derive an explicit form of the closeness estimate (22) by inserting the estimates (23) and (24) in (A.11) and (A.12), respectively. This way, by using (A.15), we derive the following explicit form of the bound (22) depending on 𝜀 and 𝑞, ‖𝑈(𝑡) − 𝑉(𝑡)‖𝓁2≤𝛼𝑡, 𝛼 = 3𝜀3+ 9𝑞𝜀2+ 12𝑞2𝜀. (36) The top and bottom rows of Fig. 5 correspond to the cases with (𝑞, 𝜔𝑏) = (0.09,0.01) and (0.34,0.46), respectively. Same as before, we employ a lattice of 𝑁= 600 sites, and use the exact KM solution of Eq. (4) as an initial condition for both the AL and DNLS models. The panels (a) and (d), and (b) and (e) showcase the spatio-temporal evolution of the amplitude of the KM solution (in the respective cases) for the AL and DNLS models, respectively, over 2 periods. In line with Fig. 4, we summarize our proximity results in panels (c) and (f) of Fig. 5 where the solid blue line (see the legend in panel (c)) shows the distance of the solutions in the 𝓁2-norm after removing the background 𝑞. The horizontal dashed black line corresponds to 𝑦= 2‖𝑈(0)‖𝓁2= 2𝜀, whereas the solid red line to 𝑦=𝛼𝑡, with 𝛼 given by (36). For panels (a)–(c), 𝜀≃ 0.054, while for panels (d)–(f) we used 𝜀≃ 0.425. 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