On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations
Abstract
The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work.
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Available online at www.sciencedirect.com ScienceDirect Journal of Differential Equations 397 (2024) 106–165 www.elsevier.com/locate/jde On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations ✩ Dirk Hennig a, Nikos I. Karachalios a, Dionyssios Mantzavinos b,∗, Jesús Cuevas-Maraver c,d, Ioannis G. Stratis e aDepartment of Mathematics, University of Thessaly, 35100, Lamia, Greece bDepartment of Mathematics, University of Kansas, Lawrence, KS 66045, USA cGrupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla, Escuela Politécnica Superior, C/ Virgen de África, 7, 41011-Sevilla, Spain dInstituto de Matemáticas de la Universidad de Sevilla (IMUS), Edificio Celestino Mutis, Avda. Reina Mercedes s/n, 41012-Sevilla, Spain eDepartment of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, GR-15784 Athens, Greece Received 4 October 2023; accepted 7 March 2024 Available online 20 March 2024 Abstract The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable ✩Acknowledgments. D.M. gratefully acknowledges support from the U.S. National Science Foundation (NSF-DMS 2206270). J.C.-M. acknowledges support from the EU (FEDER program 2014-2020) through MCIN/AEI/10.13039/501100011033 under the project PID2020-112620GB-I00. *Corresponding author. E-mail addresses: [email protected] (D. Hennig), [email protected] (N.I. Karachalios), [email protected] (D. Mantzavinos), jcuev[email protected] (J. Cuevas-Maraver), [email protected] (I.G. Stratis). https://doi.org/10.1016/j.jde.2024.03.005 0022-0396/©2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work. ©2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/). MSC: 35Q55; 37K40; 35B35 Keywords: Integrable and non-integrable focusing NLS; Power and saturable nonlinearities; Zero and nonzero boundary conditions; Structural stability; Proximity estimates; Modulational instability and soliton dynamics 1. Introduction and main results One of the most important notions concerning dynamical systems is that of structural stability, e.g. see [53], Section 3.7. Consider as an example the Cauchy (initial value) problem for the following semilinear evolution equation ut+L(u) +N(u) =0, u(x, 0)=u0(x), (1.1) along with its perturbed counterpart ut+L(u) +N(u) +f(u,x,t)=0, u(x, 0)=u0(x) +p0(x). (1.2) Here, u =u(x, t), Lis a linear differential operator and Nrepresents the nonlinearity. Furthermore, the perturbative term in (1.2)may represent external forces, dissipation terms or other effects, while p0(x) is a perturbation of the initial condition u0(x) of (1.1). The solutions of (1.1)are defined in a suitable phase space X. In the context of the structural stability theory for (1.1), the perturbations f(u, x, t)and p0(x) are small when measured in some suitable norms relevant to X. Then, the main question is whether the solution of the perturbed system (1.2) deviates too far from the solution of the original system (1.1)or not. As underlined in [53], if the evolution equation (1.1)“is very unstable, then one could doubt on its ability to accurately simulate (either numerically or theoretically) a real-life system”. On the other hand, it is also emphasized in [53] that “any equation with a good well-posedness theory is also likely 107
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 to have a good stability theory, by modifying the arguments used to prove well-posedness suitably”. In particular, stability results for (1.1) generalize the property of continuous dependence of the solution on the initial data (one of the three components of Hadamard well-posedness, the other two being existence and uniqueness), which can be obtained as a reduction of such general results in the special case of (1.2) with f=0. Similarly, the uniqueness theory of (1.1) can be approached through the special case f=p0=0. The above statements on stability theory emphasize the importance of this direction of study and motivate us to examine, in the present paper, a modified notion of stability for an evolution equation of the form (1.1)in the framework of Hamiltonian nonlinear Schrödinger equations in one spatial dimension. The role of the semilinear equation in (1.1)is assigned to the focusing cubic NLS equation iut+νuxx +μ|u|2u=0,ν,μ>0,(1.3) which is one of the fundamental integrable nonlinear dispersive partial differential equations with numerous applications in a broad range of areas within mathematical physics. Furthermore, instead of considering a “forced” counterpart of (1.3)as in (1.2), we consider focusing nonintegrable counterparts of (1.3)of the general form iUt+νU xx +γF(|U|2)U =0,ν,γ>0,(1.4) where F:R →Ris a sufficiently smooth function satisfying standard conditions that are specified later. Therefore, instead of small perturbations of (1.3), our investigation in the framework of stability is of different nature, as our primary goal is to compare systems of the same class (1.1) but with different nonlinearities: namely, the integrable NLS (1.3) with nonlinearity NI(u) = μ|u|2uagainst its non-integrable counterparts (1.4) with nonlinearities NNI(U) =γF(|U|2)U. Actually, this stability analysis is directly relevant to the proximity between the solutions of the completely integrable NLS equation (1.3) and its non-integrable counterparts (1.4). In particular, it sheds light on the potential persistence of integrable dynamics, as these are defined by the solutions and dynamical behavior of the integrable NLS equation (1.3), in the setting of the general family of non-integrable NLS equations (1.4). Studies of this generic question in the context of nonlinear lattices (discrete NLS and Ginzburg-Landau equations) are given in [35,37,36]. Nevertheless, the investigation of this question in the context of the NLS partial differential equations is a considerably more intricate task due to the following reasons: (1) Differences between the well-posedness of the integrable and the non-integrable NLS equations. The cubic NLS equation (1.3)is globally well-posed in all cases of significant boundary conditions. This is not generally the case for the non-integrable equations (1.4), for which global existence of solutions in time can be assured under smallness conditions for the initial data in suitable norms. Therefore, the investigation of stability may be carried out under either (i) the assumption of a restricted class of small initial data, in order to ensure global existence in time for the non-integrable model, or (ii) the restriction to finite times dictated by the maximal interval of existence of the non-integrable model. (2) Different boundary conditions are associated with different dynamics. The integrable NLS equation (1.3) exhibits a rich class of analytical solutions depending on the boundary conditions with which it is supplemented. The question of persistence of these analytical solutions in the dynamics of the non-integrable NLS instigates a variety of further studies in terms 108
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 of potential proximity or deviation of solutions of the integrable and non-integrable NLS, respectively. On account of the above considerations, the main result of this work is quite general and can be stated as follows. Theorem 1.1. Consider the integrable NLS equation (1.3)and the family of its non-integrable counterparts (1.4), where the nonlinearity function Fsatisfies the standard conditions |F(x)−F(y)|≤K(|x|p−1+|y|p−1)|x−y|,F(0)=0,|F(x)|≤K|x|p−1,(1.5) for some p≥1, a constant K>0and any x, y≥0. Let [0, Tmax)define a common maximal interval of existence for the solutions u(x, t)and U(x, t)to (1.3)and (1.4), respectively, associated with initial data u0(x) and U0(x) in a suitable Banach space Xgoverned by the supplemented boundary conditions for both equations. Furthermore, for each given 0 <ε<1, assume that the distance in Xbetween these initial data is of order O(ε3)while the individual norms of these data in Xare of order O(ε), i.e. u0−U0X≤Cε3,(1.6) u0X≤Cε, U0X≤Cε, (1.7) for some generic constant C>0. Then, for arbitrary finite 0 <T f<T max, there exists a constant C= C(γ, μ, ν, C, Tf)such that the corresponding solutions u(x, t) and U(x, t) satisfy the estimate sup t∈[0,Tf]u(t) −U(t)X≤ Cε3.(1.8) That is, under the assumptions (1.6)and (1.7), the distance between the solutions u(t) and U(t) measured in Xis of order O(ε3)for all t∈[0, Tf]. In view of the condition (1.6), the distance inequality (1.8) provides a generalization of the notion of continuous dependence of solutions on the initial data for the equations (1.3) and (1.4), at least for small initial data in the sense of the condition (1.7). It should be noted that the class of nonlinearities satisfying the standard conditions (1.5)is quite broad and includes, among others, the following important cases that correspond to nonintegrable NLS models: (i) The general power nonlinearity F(x) =xp, p≥1, which gives rise to the semilinear Schrödinger equation iUt+νUxx +γ|U|2pU=0.(1.9) Indeed, by the Mean Value Theorem, xp−yp=pcp−1(x −y) for some c∈(x, y), thus for any p≥1 and x, y∈Rwe have |xp−yp| =p|c|p−1|x−y| ≤p|x|p−1+|y|p−1|x−y| and the conditions (1.5)are satisfied with K=p. 109
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 (ii) The rational nonlinearity F(x) =x κ(1+x), κ>0, which corresponds to what is known as the saturable NLS equation (e.g. see [25,17,43]) iUt+νUxx +γ|U|2U 1+κ|U|2=0.(1.10) In this case, for any x, y≥0we have F(x)=1 κ(1+x)2≤1 κand |F(x)−F(y) |= |x−y| |κ|(1+x)(1+y) ≤1 κ|x−y|, thus the conditions (1.5)are satisfied with p=1 and K=1 κ. Note that the invertible transformation U(x, t) =√κe −it U(x, t) turns equation (1.10) into the alternative form i Ut+ν Uxx − U 1+ U2=0,=γ κ.(1.11) Concerning the plethora of analytical solutions to the integrable NLS equation (1.3), such as solitons, multi-soliton solutions, bound states, rational solutions and others [1,47,60], an important implication of the stability result of Theorem 1.1 is that small-amplitude localized analytical solutions of the integrable equation (1.3)persist, in the sense of the distance inequality (1.8), in the non-integrable setting of equation (1.4). More precisely, the non-integrable equation (1.4) admits small-amplitude solutions of O(ε) that stay O(ε3)close to the well-known analytical solutions of the integrable equation (1.3)for any t∈[0, Tf](note that one can impose the same initial condition u(x, 0) =U(x, 0)on both equations). This claim is further justified by the fact that, as it turns out, estimate (1.8)is also valid pointwise since it holds in L∞. A case of specific interest is the one of Tmax =∞. In this regard, we recall that, in the focusing case μ, ν, γ>0 with zero boundary conditions at infinity, global existence for the non-integrable model (1.4)is guaranteed, in general, only for small initial data. Hence, the conditions (1.7) become particularly relevant. Such conditions ensure, for example, the persistence of small-amplitude bright solitons of the integrable NLS equation (1.3)in the non-integrable setting of equation (1.4)for time intervals that can be significantly long, as well as for even more complicated dynamics like bright soliton collisions. Such long time persistence is illustrated by numerical experiments at the end of Section 2. Another application of Theorem 1.1 concerns the robustness of the persistence of analytical solutions under perturbations in the following sense: solutions that are stable in one system remain stable in the other system as long as they persist in its dynamics. This property can be proved via a transitivity argument combining the proximity estimates of Theorem 1.1 and orbital stability results for the considered solutions. The discussion leading to Corollary 2.1 below concerns standing waves. For the stability of standing waves, solitons and multi-soliton solutions, we refer the reader to the fundamental results in [15,30,31,45,46,39,57]. Importantly, the applicability of Theorem 1.1 extends beyond the framework of zero boundary conditions at infinity, as the analysis carried out in this work also concerns a broad class of nonzero boundary conditions at infinity in the general form of vanishing profiles on top of a finite nonzero background of constant amplitude. Boundary conditions of this type are of particular physical relevance. Indeed, in focusing media, nonzero boundary conditions at infinity are associated with the emergence of fascinating dynamics related to the well-known phenomenon of modulational instability. This effect, which is also known as Benjamin-Feir instability [5], 110
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 refers to the instability of a constant background to long wavelength perturbations and is ubiquitous across nonlinear science; e.g. see the review article [61], as well as the more recent works [40,51,52] that link modulational instability to the formation of rogue waves in optical media and the open sea. In the case of the integrable focusing cubic NLS equation (1.3), the nonlinear stage of modulational instability induced by the nonzero boundary conditions specified above, was studied in a series of recent works [10,11,7,8]via the inverse scattering transform and the nonlinear steepest descent method of [18]. In these works, it was rigorously shown that the solution remains bounded at all times, as one would naturally expect due to the complete integrability of the equation (1.3). Nevertheless, in the case of the general non-integrable NLS model (1.4) such integrability techniques are no longer available. In fact, in terms of global existence and regularity of solutions to the non-integrable equation (1.4), the corresponding results for nonzero boundary conditions at infinity are markedly different, e.g. see [12]as well as [13,53]. These observations motivated the study [9], which suggests the existence of a universal behavior in modulationally unstable media. In fact, by considering several non-integrable models that belong to the general NLS family (1.4), evidence is provided that they exhibit the same behavior as the one of the integrable NLS equation (1.3) established in [10,11,7,8]. The stability result of Theorem 1.1 proved in this work rigorously establishes the persistence of the nonlinear behavior that was conjectured in [9], at least at its early stages. For later times, the proximity between the dynamics depends on the type of the nonlinearity present in the nonintegrable model, as illustrated by the numerical simulations provided at the end of Section 4. This fact is highlighted by the example of a saturable nonlinearity, where the dynamics exhibits a remarkable proximity to the one of the integrable model (as in the case of zero boundary conditions). Furthermore, the numerical simulations illustrate that the smallness condition (1.7) for the initial data is by no means restrictive. In particular, they demonstrate that the modulational instability dynamics emerges from initial data that do satisfy the smallness condition in the nonintegrable case. We emphasize that the numerical findings of Section 4are predicted by our theoretical results, since Theorem 1.1 is also proved when the NLS equations (1.3) and (1.4)are considered on a finite interval and supplemented with (zero or nonzero) Dirichlet or periodic boundary conditions. Problems of this type fall under the class of initial-boundary value problems, in which the spatial domain involves an actual boundary (as opposed to extending to infinity in all directions). Such problems are significant in their own right and have been studied extensively in the literature (e.g. see [20,21] and the references therein). However, they are also directly relevant to numerical studies related to the Cauchy problem, since such studies are performed by approximating the infinite domain by a sufficiently large finite domain, supplemented with appropriate Dirichlet or periodic boundary conditions. (Some additional theoretical implications related to the question of proximity of solutions in the case of approximation by a finite domain are discussed at the end of this introductory section.) Indeed, the numerical results of Section 4are in excellent agreement with the analytical estimate (1.8). For instance, both for Dirichlet and for periodic boundary conditions, the analytical arguments establish that the growth of the distance u(t) −U(t)Xis at most linear for any t∈(0, Tmax), since u(t) −U(t)X≤Ctε3(1.12) with Cdepending only on the nonlinearity parameters γ, μbut not on t(actually, the constant C in (1.8)is given by C=CTf). We emphasize that the proof provides explicit expressions for the 111
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 dependence of the constant Con the various parameters, suggesting that these expressions can be adjusted to decrease the linear growth rate as desired. Regarding the importance of estimate (1.12), we also remark that, in a similar context, the time-growth estimates for the relevant distance function between the solutions of the complex Ginzburg-Landau and the NLS equations, when the inviscid limit of the former is considered [59], can even be exponential in time [50]. To the best of our knowledge, among the vast volume of works on NLS-type equations, the one most relevant to the present paper is [19]. However, the results of [19] concern the defocusing (as opposed to focusing) NLS equation ivt+vxx −2|v|2v=0 (1.13) supplemented with zero boundary conditions. More precisely, in [19]the authors consider nonintegrable perturbations of (1.13)in the form iVt+Vxx −2|V|2V−|V|pV=0,p>2,>0,(1.14) which correspond to μ =1, ν=−2in (1.3) and preserve the defocusing nature of (1.13). The main result of [19] can be outlined as follows: for sufficiently smooth initial data that decay at an appropriate rate (described by a suitably defined weighted Sobolev space) and small >0, the solutions of (1.14) approach those of (1.13)as t→∞in the sense of the estimate v(t) −V(t)L∞(R)=O1 t1 2+κ,κ>0.(1.15) This result is proved by combining the inverse scattering transform method (and, in particular, by studying the behavior of the reflection coefficient that emerges in the integrable case (1.13)) with detailed estimates in appropriate Sobolev norms derived for the perturbed (non-integrable) model (1.14). The model (1.14) considered in [19] falls under the general perturbative framework (1.2). On the other hand, as noted earlier, the results presented herein are of different nature, since now the integrable model (1.3)is compared against its distinct non-integrable counterparts (1.4) which, unlike (1.14), cannot be treated as weak perturbations. Another major difference between [19] and the present work stems from the fact that, for zero boundary conditions, while the solutions of the defocusing models (1.13) and (1.14)are known to exist globally in time for all initial data and to decay for a certain range of nonlinearity exponents [32]even in higher than one spatial dimensions [33,34], the solutions of the non-integrable focusing NLS equation (1.4) are globally defined only for small initial data (see Theorem 2.1 below). Thus, our proximity estimates between the solutions of two essentially different systems, namely (1.3) and (1.4), concern in general finite time intervals in the spirit of the continuous dependence of solutions on the associated small initial data, and do not explore the long-time asymptotic behavior considered in [19]. In addition, here we also consider the important case of nonzero boundary conditions at infinity, which was not investigated in [19]. For the proof of Theorem 1.1, we employ the Fourier transform for the evolution equation satisfied by the difference of solutions (t) =u(t) −U(t). We also remark on an alternative approach through energy estimates and interpolation inequalities. This second approach is applicable in all the cases of boundary conditions, albeit with distinct implications for each specific case. For the case of zero boundary conditions, we take advantage of the global existence results 112
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 for the non-integrable NLS equation in the case of small initial data and also of the regularity of solutions when this data belong to a suitable class. The case of nonzero boundary conditions at infinity turns out to be more challenging since, even for small initial data, it is not known in general whether the non-integrable focusing NLS equation is globally well-posed [3,49]. Indeed, global existence is only guaranteed in the defocusing case [24,26,29,28]. On the real line, we prove local existence in H1(R)and then use a continuity argument for small data in the sense of (1.7)in order to establish a closeness estimate between the solutions of the integrable and non-integrable systems, at least for a short time period. Furthermore, motivated by the numerical results of [9], which indicate the universality of the modulational instability dynamics beyond the integrability barrier, we analytically study the finite interval problem for equations (1.3) and (1.4) supplemented with the relevant nonzero Dirichlet boundary conditions. This finite domain problem is different from the problem on the real line in that, although both problems possess a conservation law for the L2norm that involves the amplitude q0of the wave background, global existence at the L2level can be deduced from that law only in the case of the finite domain. Hence, up to the critical nonlinearity p=2, we prove (see Theorem 4.1 below) that the solutions to the non-integrable NLS equation (1.4)on the finite interval (−L, L) exist globally in time for appropriate smallness conditions on Land the initial data (in the subcritical case 1 ≤p<2, the latter condition is not necessary). It should be noted that the upper bound on Ltends to infinity as the amplitude q0of the nonzero background tends to zero, i.e. when the problem approaches the one with zero boundary conditions. Moreover, the proximity estimates of Theorem 1.1 under the conditions (1.6) and for q0=O(ε) are valid for L =O(1/ε). Therefore, the accuracy of the closeness estimate between the solutions of the integrable and non-integrable NLS equations improves when the relevant norms are evaluated over the interval (−1/ε, 1/ε) around the core of the respective modulational instability pattern. This fact is further discussed in Section 4, where it is also illustrated numerically by the simulations of Figs. 4.3 and 4.5. Finally, we comment on the case of periodic boundary conditions, which can also be used for approximating the problem on the real line supplemented with zero or nonzero boundary conditions. The proofs in the periodic case are similar to the ones in the finite interval case. Note, in particular, that the assumptions for global existence of solutions to the non-integrable model are also similar, as the main conservation laws for the energy and power are the standard ones. Regarding numerical simulations, however, if the periodic problem is used in order to approximate the infinite line problem with nonzero boundary conditions by taking the parameter Lto be large, then a complication arises: the periodic boundary conditions define problems with finite energies in Sobolev spaces of periodic functions [4], which is not the case for the problem on the real line. Thus, finite domain approximations may not capture essential effects and implications associated with the infinite domain problem. This issue is especially highlighted by the results of the present paper through the contrast between the local existence for the problem on the real line and the global existence for the finite domain approximation with nonzero Dirichlet boundary conditions. Structure of the paper. In Section 2, we prove Theorem 1.1 in the case of vanishing boundary conditions, namely we establish Theorem 2.2. In addition, we present numerical studies for the concrete examples of bright solitons and soliton collisions. In Section 3, we establish the version of Theorem 1.1 associated with the case of non-vanishing boundary conditions emerging from a constant nonzero background, namely Theorem 3.2. A key result, which is used for the derivation of the proximity estimates, is the proof of Theorem 3.1 for the local existence of solutions to the 113
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 non-integrable NLS models in the class C([0, T]; H1(R)). In Section 4, we turn our attention to the case of a finite domain. First, we prove the version of Theorem 1.1 for the problem on a finite interval with nonzero Dirichlet boundary conditions, namely Theorem 4.2, then we establish the corresponding result (Theorem 4.3) for periodic boundary conditions, and finally we conclude with a numerical study simulating the problem with nonzero boundary conditions for a variety of localized initial data on the top of a finite background. 2. Zero boundary conditions at infinity In this section, we establish the version of Theorem 1.1 that corresponds to the scenario in which both the integrable NLS equation (1.3) and its non-integrable counterpart (1.4) satisfy zero boundary conditions at infinity, namely lim |x|→∞u(x, t) =lim |x|→∞U(x,t)=0,t≥0.(2.1) We begin by recalling the following well-known global existence and regularity results for the general non-integrable NLS equation (1.4). Theorem 2.1 (Existing results on well-posedness). Consider the Cauchy problem on the line for the non-integrable NLS equation (1.4)with zero conditions at infinity. (i) If 1 ≤p<2, then for any initial datum U0∈H1(R)there exists a global in time solution U(x, t) which is uniformly bounded in H1(R), i.e. there is a constant M>0independent of tsuch that sup t≥0U(t)H1(R)≤MU0H1(R).(2.2) Furthermore, if U0∈Hm(R)with m >1then the solution exists globally in Hm(R)and is uniformly bounded in Hm(R). (ii) If p≥2, then there exists a constant δ>0such that if U0H1(R)≤δthen the solution U(x, t) exists globally in time and is uniformly bounded in H1(R), satisfying the estimate (2.2)for some constant M>0. Furthermore, for initial data U0∈Hm(R)with m >1 the solution possesses the same regularity as in case (i). For the proofs of the results stated in Theorem 2.1, we refer the reader to the monographs [12,13,53]. Specifically for the further regularity properties of solutions, we also refer to [32,33]. We now proceed to the main result of this section, which provides the counterpart of Theorem 1.1 in the case of zero boundary conditions at infinity. Theorem 2.2 (Theorem 1.1 for zero boundary conditions at infinity). Let p≥1and suppose that the integrable and non-integrable focusing NLS equations (1.3)and (1.4)are supplemented with the initial conditions u(x, 0) =u0(x) and U(x, 0) =U0(x), respectively. (i) L2closeness: Given 0 <ε<1, suppose that the initial data satisfy u0−U0L2(R)≤Cε3,(2.3) 114
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 U(t)L2(R)= 2−p 2pU(2t)L2(R)(2.36) and so for p<2 one may construct standing waves of arbitrary L2norm for a suitable choice of . Moreover, for X:=x we have ∂xU(x, t) = p+1 p∂XU(X, 2t) and so ∂xU(t)L2(R)= p+2 2p∂xU(2t)L2(R).(2.37) Therefore, the scaling (2.35) preserves the stability of standing waves. Indeed, suppose W∈ Apcorresponds to a stable standing wave. Then, given >0 and initial data U0satisfying the hypothesis of Theorem 2.3, for W(x) :=1 pW(x) and U0(x) defined analogously we have, in view of (2.36) and (2.37), U0−WH1(R)≤max 2−p 2p, 2+p 2pδ(2.38) and sup t>0 inf θ∈Rinf y∈RU(·,2t)−eiθW(·−y)H1(R)≤max 2−p 2p, 2+p 2p, (2.39) implying the stability of the standing wave associated with W. Note that the restriction p<2 ensures that the exponents in (2.36) and (2.37) can both be controlled by sufficiently small; otherwise, if p>2 then the former equality requires large and the latter small, so they cannot be combined to control the H1norm. With the above preparations, a transitivity argument that combines Theorem 2.2 proved in this work with Theorem 2.3 from [13] implies the following result. Corollary 2.1. Suppose 1 <p<2and let UNI(x, t) =eiωt WNI(x), WNI ∈Ap, ω>0, be a standing wave solution of the non-integrable NLS equation (1.9)which is stable in the sense of Theorem 2.3. Furthermore, given 0 <ε<1, let u0, U0∈H1(R)be initial data for the integrable and non-integrable NLS equations (1.3)and (1.9), respectively, satisfying the conditions U0−WNIH1(R)≤δ(ε), (2.40) u0H1(R)≤c0ε, (2.41) u0−U0H1(R)≤c1ε, (2.42) for some constants c0, c1>0, with δ(ε) satisfying the stability criterion of Theorem 2.3. Then, u0−WNIH1(R)≤c1ε+δ(ε) =: δ(ε) and for arbitrary 0 <T f<∞there exists a constant K(Tf) >0such that sup t∈[0,Tf] inf θ∈Rinf y∈Ru(·,t)−eiθ(t)WNI(·−y(t))H1(R)≤ Kε. (2.43) Proof. Let 0 <ε<1. First, note that the conditions (2.40) and (2.42) combined with the triangle inequality readily yield u0−WNIH1(R)≤u0−U0H1(R)+U0−WNIH1(R)≤ δ(ε). Next, 121
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 since WNI and U0satisfy the hypothesis of Theorem 2.3, the solution Uof the non-integrable NLS equation (1.9) associated with U0satisfies the bound sup t>0 inf θ∈Rinf y∈RU(·,t)−eiθWNI(·−y)H1(R)≤ε. (2.44) Let udenote the solution of the integrable NLS equation (1.3) associated with u0. Then, in view of the conditions (2.41) and (2.42), which via the triangle inequality and the fact that ε<1imply U0H1(R)≤(c0+c1)ε≤(c0+c1)ε1 3, estimate (2.7)of Theorem 2.2 yields u(t) −U(t)H1(R)≤ C1ε, t ∈[0,T f], C1= C1(Tf). (2.45) Hence, combining the triangle inequality with (2.44) and (2.45), we deduce sup t∈[0,Tf] inf θ∈Rinf y∈Ru(·,t)−eiθWNI(·−y)H1(R) ≤sup t∈[0,Tf]u(t) −U(t)H1(R)+sup t>0 inf θ∈Rinf y∈RU(·,t)−eiθWNI(·−y)H1(R)≤ C1ε+ε, which amounts to the desired estimate (2.43) with K= C1+1. Numerical simulations. We conclude this section with a numerical study illustrating our analytical results in the case of the zero boundary conditions (2.1). As a first example, we consider the case of bright solitons. The second example concerns the more intricate case of collision of bright solitons. Bright solitons. We supplement the non-integrable NLS equation (1.4) with the initial condition emanating from the one-soliton solution of the integrable NLS equation (1.3) (e.g. see [41]), namely U0(x) =uS(x, 0):=Asech(Ax)eicsx(2.46) with the parameters A, cschosen so that the smallness conditions of Theorem 2.2 are met. More specifically, we trigger the dynamics of the non-integrable NLS equation (1.4) with ν=1/2, γ=1 and the initial condition (2.46) with small amplitude A =0.02 and velocity cs=1. We examine the case of a power nonlinearity (1.9) with p=3(top left panel in Fig. 2.1) as well as the saturable nonlinearity model (1.10)(top right panel in Fig. 2.1). As explained below, in both cases the evolution is essentially indistinguishable from the one of the integrable NLS equation. The bottom panels of Fig. 2.1 show the evolution of the distance norms L2(R) (middle (blue) line), H1(R)(upper (yellow) line) and L∞(R)(bottom (red) line). The evolution of these norms appears to be in an excellent agreement with the theoretical prediction (2.24)for their linear growth with respect to time. For the aforementioned set of parameter values, we have uS(0)H1(R)=U0H1(R)≃0.2, i.e. we may take ε=0.2in which case there is an excellent agreement with the theoretical predictions regarding the growth of the distance in terms of ε. Indeed, for times t∼O1/ε2the bound (2.24) predicts that (t)X∼O(ε), which is consistent with the evolution of the distance norms over the interval t∈[0, Tf]with Tf=100 depicted in the bottom row of Fig. 2.1. 122
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 2.1. Top row: Spatiotemporal evolution of the density |U(x, t)|2of the non-integrable NLS equation (1.4) with ν=1/2, γ=1and supplemented with the soliton initial condition (2.46). Left: Power nonlinearity (1.9) with p=3(supercritical case). Right: Saturable nonlinearity (1.10). Bottom row: The corresponding evolution of the norms (t)X with X=L2, H1or L∞for each of the two cases. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) Note that in the case of the saturable nonlinearity the linear growth of (t)Xis incremental, suggesting that the dynamics of the saturable NLS equation may be even closer to those of the integrable one, compared to the power nonlinearity model. This behavior also insinuates that in the non-integrable cases the persistence of small-amplitude bright solitons may last for a significantly long time, which is quite remarkable. The left panel of Fig. 2.2 depicts the numerical evolution of the center of mass of the soliton, presented by solid lines for the non-integrable models and by a dashed line for the integrable model (in fact, the two solid lines for the power and saturable nonlinearities are indistinguishable). For this study, we use a spatial translation of the initial condition (2.46), namely U0(x) =Asech[A(x −x0)]eics(x−x0)with x0=125 and A =0.02, cs=1as before. The two lines corresponding to the non-integrable models are indistinguishable from the one of the integrable NLS which is extended in the figure also to x<0for visualization purposes. The solitary waves of the non-integrable models demonstrate remarkable robustness. Besides the case of small-amplitude initial data considered above, it is also important to investigate the case when the value of the parameter εis increased. The right panel of Fig. 2.2 shows logarithmic scale plots of the variation of the distance norm (t)Xfor X=L2(R), H1(R)or L∞(R)as a function of εfor fixed Tf=600. The dots along the solid lines correspond to the numerically obtained rates of these variations fitted to the lines of the form 123
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 2.2. Left: Spatiotemporal evolution of the soliton’s center of mass for the integrable and the non-integrable models. Right: Logarithmic scale plots of the variation of the distance norm (t)Xwith X=L2, H1or L∞as a function of εfor fixed Tf=600. See text for more details. (Tf)L2(R)versus K1εσ1, (Tf)H1(R)versus K2εσ2, and (Tf)L∞(R)versus K3εσ3. The fitting results in the following values: K1=8.87, K2=17.77, K3=2.96 and σ1=σ2=4.5, σ3=4.86. For these values of constants Kj, the dashed lines correspond to the analytical estimates of Theorem 2.2 of the form (Tf)Xversus Kjε3. The numerical results are consistent with the order of the analytical estimates and, in fact, indicate that the numerical variation of the distance norms may be of significantly lower rate, namely of O(ε4.5)or O(ε5). Bright soliton collisions. For the study of the dynamics of bright soliton collisions, we choose as initial datum two incoming soliton solutions of the cubic NLS equation (1.3)of the same amplitude A, initially separated by a distance x0and moving against one other at speed cs, namely U0(x) =Asech [A(x −x0)]e−ics(x−x0)+Asech [A(x +x0)]eics(x+x0).(2.47) We take A =0.1 and x0=50. For the velocities, we considered both fast solitons with cs=2 and very slow solitons with cs=0.02. Fig. 2.3 depicts the results for the soliton collision dynamics in the case of the fast solitons. The top row corresponds to the power nonlinearity model (1.9)in the supercritical case p=3 and the bottom row to the saturable NLS equation (1.10). The plots of the spatiotemporal evolution of the density |U(x, t)|2in the left panels reveal that the non-integrable systems exhibit collision dynamics that are almost identical to the integrable (cubic) case. This persistence of integrable soliton dynamics is even more evident in the plots of the evolution of the distance norm (t)X with X=L2(R), H1(R)or L∞(R), which are in excellent agreement with the predictions of Theorem 2.2. Note that the collision event is captured by the evolution of norms as the sigmoidshaped portion of the curve featuring an interlude of increased, but still moderate, slope. Furthermore, the evolution of norms highlights two features: (i) The saturable NLS equation seems to be a more structurally stable non-integrable model with respect to the dynamics of the integrable NLS equation, an effect which was already observed in the dynamics of the single bright soliton earlier. 124
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 2.3. Fast bright soliton collisions. Left column: Spatiotemporal evolution of the density |U(x, t)|2initiated by the datum (2.47) with A =0.1, x0=50, cs=2for the non-integrable NLS equation with power nonlinearity (1.9)in the supercritical case p=3 (top) and the non-integrable saturable NLS equation (1.10) (bottom). Right column: Evolution of the norms (t)Xwith X=L2, H1or L∞for the power nonlinearity p=3 (top) and the saturable nonlinearity (bottom). In both cases, ν=1/2and μ =γ=1. (ii) The relatively slow growth of (t)L∞(R)indicates that the solutions are closer in the sense of pointwise convergence than they are in the sense of convergence in the L2(R) or H1(R)norms; a larger deviation in the latter norms hints at rather complex dynamics on smaller scales that may contribute to the overall difference between the integrable and non-integrable cases. These two features are further elucidated by the study of the slow solitons collisions shown in Fig. 2.4. There, the top row corresponds to the power nonlinearity with p=3. The breakdown of integrability in this case appears to be dramatic, as it is illustrated by the spatiotemporal evolution of the density in the top left panel. Part of the energy is trapped at the collision site, a common feature also present in other non-integrable models (e.g. the Klein-Gordon equation φ4). This is not the case for the saturable nonlinearity, as shown in the bottom row. Qualitatively, the dynamics of the saturable NLS equation is almost identical to the one of the integrable NLS equation and, in contrast to the power nonlinearity, no energy trapping effect is observed. The vast difference between the two non-integrable models is also prominent in the corresponding evolution of the distance norms (t)X. For the septic power nonlinearity, the deviation of 125
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 2.4. Slow bright soliton collisions. Same as in Fig. 2.3 but with cs=0.02. the L∞norm is still moderate and compliant with an L∞closeness of solutions. However, the larger growth and the behavior of the L2and H1norms (particularly after the collision) gives further demonstration of the integrability breakdown induced by the power nonlinearity. On the other hand, the corresponding study of the behavior of (t)Xfor all three norms in the case of the saturable NLS equation provides additional evidence of structural stability of that model in reference to the integrable one. 3. Nonzero boundary conditions at infinity The main objective of this section is to prove the analogue of Theorem 1.1 — namely, Theorem 3.2 below — on the proximity of solutions between the integrable and non-integrable NLS equations (1.3) and (1.4) when these are supplemented with a broad class of nonzero boundary conditions at infinity. For this task, we first assume (local) existence of solutions to the nonintegrable problem in H1(R)in order to derive the relevant closeness estimates. In this regard, we note that the well-posedness results for the general non-integrable problem with nonzero boundary conditions are quite limited when compared to the plethora of results in the case of zero boundary conditions and, in particular, the powerful results of Theorem 2.1. Thus, en route to establishing Theorem 3.2, we also prove local existence in H1(R)for the non-integrable NLS equations (1.9) and (1.10) that correspond to the power and saturable nonlinearities, respectively, and as such represent the two most prominent members of the general family of equations (1.4). 126
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Motivated by the physically significant description given in [11,9,28], we supplement the integrable and non-integrable NLS equations (1.3) and (1.4) with the following general class of nonzero boundary conditions at infinity: lim |x|→∞u(x, t) =lim |x|→∞eiμq2 0tζ(x), lim |x|→∞U(x,t)=lim |x|→∞eiγF(q2 0)t ζ(x), t ≥0,(3.1) where the complex-valued function ζ(x) belongs to the Zhidkov space X1(R):=ζ∈L∞(R):ζ∈L2(R)(3.2) and satisfies lim x→±∞ζ(x)=ζ±∈C,|ζ±|=q0>0.(3.3) According to the above boundary conditions, the behavior of the solutions u(x, t) and U(x, t) as |x| →∞is dictated by the function ζ(x), which approaches a constant background of size q0>0as described by the limit conditions (3.3). The precise form of the conditions (3.1) can be motivated as follows. Let u(x, t) and U(x, t) be solutions of equations (1.3) and (1.4), respectively, such that lim x→±∞u(x, t) =u±(t), lim x→±∞U(x,t)=U±(t), (3.4) where U±(t) and u±(t) are temporal functions of constant modulus equal to q0but are otherwise to be determined. Then, taking the limit of (1.3) and (1.4)as x→±∞while assuming that the nonlinearity function Fhas sufficient smoothness so that lim|x|→∞ F(|U|2) = F(lim|x|→∞ |U|2) =F(q2 0)(note, in particular, that this is true both in the semilinear case (1.9) and in the saturable case (1.10)), we have i(u±)t+μq2 0u±=0⇒u±(t) =eiμq2 0tu±(0), i(U±)t+γF(q2 0)U±=0⇒U±(t) =eiγF(q2 0)t U±(0). (3.5) Therefore, if the initial data are such that limx→±∞ u(x, 0) =limx→±∞ U(x, 0) =ζ±, i.e. if u±(0) =U±(0) =ζ±, then for ζsatisfying (3.3)the expressions (3.4) and (3.5)give rise to the conditions (3.1). A few remarks are now in place: (i) The assumption for ζ∈X1(R)covers the standard case ζ(x) =ζ0∈Cbeing a constant with |ζ0| =q0>0. (ii) The pure step function ζs(x) =ζ+∈C,x>0, ζ−∈C,x<0, does not belong to X1(R), since ζ s(x) =(ζ+−ζ−)δ(x) in the sense of distributions and so ζ s/∈L2(R). Nevertheless, although our analysis does not cover the pure step function ζs, it 127
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 does cover any continuous function approximating ζs, as the derivative of such a function will belong to L2(R)(and hence that function will belong to X1(R)). (iii) Generalizations from the pure step function ζsto a nontrivial ζbelonging to the class (3.2)-(3.3) covered by the present work are relevant in the context of the dynamics of NLS equations. In the case where q0=0, localized waveforms in the form of Peregrine solitons may locally emerge on a nontrivial decaying ζ(x) induced by the initial conditions for the semiclassical limit of the focusing integrable NLS equation [6]. For fundamental results on the semiclassical problem for NLS, we refer to [38,48,56]. The local emergence of localized structures reminiscent of the Peregrine soliton has been observed also in experimental setups [54,55] and is expected to be robust in the presence of damping and forcing effects [22,23]. However, herein we will restrict ourselves to the case q0>0. The boundary conditions (3.1) can be made time independent (i.e. constant) via the change of variables u(x, t) =eiμq2 0tq(x,t), U(x,t) =eiγF(q2 0)t Q(x, t). (3.6) Then, equations (1.3) and (1.4)take the form iqt+νqxx +μ|q|2−q2 0q=0,(3.7) iQt+νQxx +γF(|Q|2)−F(q2 0)Q=0,(3.8) the initial conditions remain unchanged, namely q(x, 0) =u0(x) and Q(x, 0) =U0(x), and the boundary conditions become lim x→±∞q(x,t) =lim x→±∞Q(x, t) =ζ±(3.9) so that lim|x|→∞ |q(x, t)| =lim|x|→∞ |Q(x, t)| =q0>0. Furthermore, the additional change of variables q(x,t) =φ(x,t)+ζ(x), Q(x,t)=(x, t) +ζ(x) (3.10) leads to the modified NLS equations iφt+ν(φ+ζ)xx +μ|φ+ζ|2−q2 0(φ +ζ)=0,(3.11) it+ν(+ζ)xx +γF(|+ζ|2)−F(q2 0)( +ζ)=0,(3.12) along with the initial conditions φ(x,0)=u0(x) −ζ(x)=:φ0(x), (x, 0)=U0(x) −ζ(x)=:0(x) (3.13) and zero boundary conditions at infinity, i.e. lim |x|→∞φ(x,t)=lim |x|→∞(x, t) =0,t≥0.(3.14) 128
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 The rest of this section is organized as follows. First, we establish L2and H1closeness estimates for the modified NLS equations (3.11) and (3.12). Through the transformations (3.6) and (3.10), these closeness results imply corresponding estimates for the difference e−iμq2 0tu(x, t) −e−iγF(q2 0)t U(x,t) involving the solutions of the original integrable and non-integrable NLS equations (1.3) and (1.4). The derivation of the closeness estimates is accomplished under the assumption of (at least) local H1existence for the non-integrable equation (1.4). In this connection, we emphasize that the case of nonzero boundary conditions is substantially different from the one of zero boundary conditions, even at the fundamental level of well-posedness. In particular, to the best of our knowledge, the well-posedness of the non-integrable NLS equation (1.4) with general nonlinearities is much less understood in the former case. In the focusing regime, one of the few known results in Sobolev spaces is due to [49], where local existence was established for the semilinear Schrödinger equation (1.9)in the case of Hs(R)perturbations of the background wave with s>1/2. It should be noted that the proofs in [49]are given only for p=1 corresponding to the integrable cubic NLS equation, although it is remarked that the arguments should go through also in the case of general p>1. In this regard, below we also prove local existence in H1(R)for the modified NLS equation (3.12) (and hence for the non-integrable NLS equation (1.4)) in the cases of a general power nonlinearity as well as of a saturable nonlinearity, which are respectively associated with the models (1.9) and (1.10). This local existence result is crucial as (i) it removes the relevant assumption used for the derivation of the closeness estimates, and (ii) via a continuity argument with respect to time (since our local solutions end up in the class C([0, T]; H1(R))), it allows us to obtain the analogue of Theorem 1.1 in the case of the nonzero boundary conditions (3.1), namely Theorem 3.2. Closeness estimates. As noted above, in order to investigate the proximity between the solutions of equations (3.11) and (3.12), we first assume local existence of these solutions in H1(R). Local estimates validating this assumption in the case of a general power nonlinearity and the saturable nonlinearity are obtained after the proof of the closeness estimates. The difference of solutions to equations (3.11) and (3.12)is the same as the one of solutions to equations (3.7) and (3.8), because it is unaffected by the change of variables (3.10). On the other hand, the change of variables (3.6) does not preserve that difference, as it results in different phase factors multiplying each of the solutions to the original equations (1.3) and (1.4). That is, (x, t) :=φ(x,t)−(x, t) =q(x,t)−Q(x, t) =e−iμq2 0tu(x, t) −e−iγF(q2 0)t U(x,t) with satisfying the equation i t+ν xx =γG F,(x, t) −μG 1,φ(x, t), (3.15) where GF,(x, t) =F(|+ζ|2)−F(q2 0)( +ζ) (3.16) and the subscript “1” denotes the identity function, so that G1,φ(x, t) =|φ+ζ|2−q2 0(φ +ζ). Starting from equation (3.15), we first establish closeness estimates in L2(R)and then in H1(R), the latter also implying the result in L∞(R)via Sobolev embedding. 129
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 L2closeness. Taking the Fourier transform (2.11)of equation (3.15) and integrating over t, we find (ξ, t) =e−iνξ2t (ξ, 0)−i t 0 e−iνξ2(t−τ) γ GF,(ξ, τ) −μ G1,φ(ξ, τ)dτ. (3.17) Hence, by Plancherel’s theorem, Minkowski’s integral inequality and the triangle inequality, (t)L2(R)≤1 2π (0)L2(R) +1 2π t 0e−iνξ2(t−τ) γ GF,(ξ, τ) −μ G1,φ(ξ, τ)L2(R)dτ = (0)L2(R)+ t 0γGF,(τ)L2(R)+μG1,φ(τ)L2(R)dτ (3.18) and we need to estimate the spatial L2norms of GF, and G1,φ . Note that the conditions (3.3) and (3.14) already imply lim|x|→∞ GF,(x, t) =lim|x|→∞ G1,φ(x, t) =0. More precisely, we have the following estimate: Lemma 3.1. Let Fsatisfy the properties (1.5). Then, for any p≥1and each t≥0, the function GF, defined by (3.16)admits the bound GF,(t)L2(R) ≤2√2K(t)L∞(R)+ζL∞(R)+q02p(t)L2(R)+|ζ|−q0L2(R)(3.19) where Kis the constant associated with Fvia (1.5). Proof. By the first of the properties (1.5), we have F(|+ζ|2)−F(q2 0)≤K|+ζ|2(p−1)+q2(p−1) 0|+ζ|2−q2 0,p≥1. Then, noting that |+ζ|2−q2 0≤(||+|ζ|)2−q2 0,|+ζ|≥q0, q2 0−|||−|ζ||2,|+ζ|≤q0, we obtain |+ζ|2−q2 0≤(||+|ζ|)2−q2 0+(||−|ζ|)2−q2 0 ≤2(||+|ζ|+q0)||+|ζ|−q0(3.20) 130
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 In turn, we can estimate the difference of nonlinearities in L2as follows: GF, −GF,L2(R) ≤K+ζ2(p−1) L∞(R)++ζ2(p−1) L∞(R)+ζL∞(R)++ζL∞(R) ·+ζL∞(R)++ζ2p L∞(R)+q2p 0−L2(R) ≤K+ζL∞(R)++ζL∞(R)2p+q2p 0−L2(R) ≤KL∞(R)+L∞(R)+2ζL∞(R)+q02p−L2(R) ≤KH1(R)+H1(R)+2ζL∞(R)+q02p−L2(R)(3.38) with the last step due to Sobolev embedding. It remains to also estimate the L2norm of the derivative ∂xGF, −GF,. This task turns out to be more involved due to the appearance of the difference F(| +ζ|2) −F(| +ζ|2). First, differentiating (3.36) and rearranging appropriately, we have ∂xGF, −GF,=F(|+ζ|2)(+ζ)x+ζ+(+ζ)+ζx −F(|+ζ|2)(+ζ)x+ζ+(+ζ)+ζx( +ζ) +F( |+ζ|2)−F( |+ζ|2)( +ζ)x +F(|+ζ|2)(+ζ)x+ζ+(+ζ)+ζx( −) +F( |+ζ|2)−F(q2 0)( −)x.(3.39) Therefore, using also the conditions (1.5), the embedding H1(R) ⊂L∞(R)and inequality (3.37), we obtain ∂xGF, −GF,L2(R)≤F(|+ζ|2)(+ζ)x+ζ+(+ζ)+ζx −F(|+ζ|2)(+ζ)x+ζ+(+ζ)+ζxL2(R) ·H1(R)+ζL∞(R) +22p−1KH1(R)+H1(R)+ζL∞(R)2p−1 ·H1(R)+ζL2(R)−H1(R) +2KH1(R)+ζL∞(R)2p−1H1(R)+ζL2(R)−H1(R) +KH1(R)+ζL∞(R)+q02p−H1(R).(3.40) The difference −H1(R)has conveniently appeared in the last three terms on the right-hand side of the above inequality. In order to also extract it from the first term, we use the identity 137
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 u1v1w1−u2v2w2=u1v1w1−w2+u1w2(v1−v2)+v2w2(u1−u2) to express the first of the differences involved in the first term of (3.40)as F(|+ζ|2)(+ζ)x+ζ−F(|+ζ|2)(+ζ)x+ζ =F(|+ζ|2)(+ζ)x−+F(|+ζ|2)+ζ(−)x +(+ζ)x+ζF(|+ζ|2)−F(|+ζ|2). In view of this writing and also the properties (1.5) satisfied by F, we find F(|+ζ|2)(+ζ)x+ζ−F(|+ζ|2)(+ζ)x+ζL2(R) ≤F(|+ζ|2)L∞(R)x+ζL2(R)−L∞(R)+F(|+ζ|2)L∞(R)+ζL∞(R) ·(−)xL2(R)+(+ζ)xL2(R)+ζL∞(R)F(|+ζ|2)−F(|+ζ|2)L∞(R) ≤KH1(R)+ζL∞(R)2(p−1)H1(R)+H1(R)+ζL2(R)+ζL∞(R)−H1(R) +H1(R)+ζL2(R)H1(R)+ζL∞(R)F(|+ζ|2)−F(|+ζ|2)L∞(R).(3.41) Furthermore, by symmetry with respect to complex conjugation, the exact same bound is also satisfied by the L2norm of the second difference in the first term of (3.40), i.e. F(|+ζ|2)(+ζ)+ζx−F(|+ζ|2)(+ζ)+ζxL2(R)≤RHS(3.41).(3.42) Thanks to the bounds (3.41) and (3.42), it suffices to extract the H1norm of −from the norm F(|+ζ|2)−F(|+ζ|2)L∞(R).(3.43) This norm cannot be handled as easily as the L∞norm of F( |+ζ|2) −F( |+ζ|2)that arose earlier in the L2estimation of (3.36). Indeed, note in particular that the first of the conditions (1.5) is not necessarily satisfied by F. For example, in the case of power nonlinearity F(x) =xpwe have F(x) =pxp−1and, unless p≥3 2, it is not possible to bound the difference |F(x) −F(y)| by a factor of √x−√y(let alone |x−y|), which is the minimum requirement in order to make −L∞(R)appear in the bound for (3.43). On the other hand, in the case of the saturable nonlinearity F(x) =x κ(1+x) we have F(x) −F(y)=1 |κ| 1 (1+x)2−1 (1+y)2=|2+x+y| |κ|(1+x)2(1+y)2|x−y| 1 |κ|1 1+x3+1 1+y3|x−y|≤2 |κ||x−y|,x,y≥0, (3.44) and so it seems possible to control (3.43) by −L∞(R). 138
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Following the above observations, we revisit the estimation of ∂xGF, −GF,L2(R)by treating the two cases F(x) =x κ(1+x) (saturable nonlinearity) and F(x) =xp(power nonlinearity) separately. Contraction estimate for the saturable nonlinearity. In this case, F(x) =x κ(1+x) and F(x) = 1 κ(1+x)2are uniformly bounded by 1 |κ|for x≥0. In addition, the first of the conditions (1.5)is met with K=1 |κ|and p=1. Thus, returning to (3.39) and using these uniform bounds along with the embedding H1(R) ⊂L∞(R)and inequality (3.37)as appropriate, we have ∂xGF, −GF,L2(R) ≤F(|+ζ|2)(+ζ)x+ζ+(+ζ)+ζx −F(|+ζ|2)(+ζ)x+ζ+(+ζ)+ζxL2(R)H1(R)+ζL∞(R) +2 |κ|H1(R)+H1(R)+ζL∞(R)−H1(R)H1(R)+ζL2(R) +2 |κ|H1(R)+ζL∞(R)H1(R)+ζL2(R)−H1(R) +2 |κ|−H1(R).(3.45) Notice that the bound (3.44) together with inequality (3.37)imply F(|+ζ|2)−F(|+ζ|2)≤2 |κ||+ζ|+|+ζ||−|. Hence, in the saturable case, the norm (3.43) satisfies the inequality F(|+ζ|2)−F(|+ζ|2)L∞(R) ≤4 |κ|H1(R)+H1(R)+ζL∞(R)−H1(R), which can be combined with (3.41) and the fact that p=1 and K=1 |κ|to deduce F(|+ζ|2)(+ζ)x+ζ−F(|+ζ|2)(+ζ)x+ζL2(R) ≤1 |κ|H1(R)+H1(R)+ζL2(R)+ζL∞(R)+4H1(R)+ζL2(R) ·H1(R)+ζL∞(R)H1(R)+H1(R)+ζL∞(R)−H1(R). Moreover, by symmetry, the same estimate also holds for the left-hand side of (3.42). Therefore, returning to (3.45), we eventually obtain 139
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 ∂xGF, −GF,L2(R) ≤8 |κ|1+H1(R)+H1(R)+ζL2(R)+ζL∞(R)4−H1(R). (3.46) The L2estimates (3.38) and (3.46) combine to yield the H1estimate GF, −GF,H1(R) ≤16 |κ|1+H1(R)+H1(R)+ζL2(R)+ζL∞(R)+q04−H1(R) which in turn implies, via inequality (3.35), [](t) −[](t)H1(R)≤32γ |κ|tsup τ∈[0,t]1+(τ)H1(R)+(τ)H1(R)+ζL2(R) +ζL∞(R)+q04( −)(τ)H1(R). Therefore, a sufficient condition for the map →[]to be a contraction in the ball B(0, ρ) ⊂ C([0, Tf]; H1(R)) is Tf<|κ| 32γ1+2ρ+ζL2(R)+ζL∞(R)+q04.(3.47) Observe that (3.47)is a stronger condition than (3.34). Contraction estimate for the power nonlinearity. By the triangle inequality, ∂xGF, −GF,L2(R)≤∂x|+ζ|2p( +ζ)−|+ζ|2p( +ζ)L2(R) +|q0|2p−H1(R).(3.48) We then invoke the following widely used result, whose proof we include below for completeness. Lemma 3.3. For any p≥1and any pair of complex numbers z, z, |z|2pz−|z|2pz=(p +1) 1 0 |Zλ|2pdλ!z−z+p 1 0 |Zλ|2(p−1)Z2 λdλ!(z−z),(3.49) where Zλ:=λz +(1−λ)z, λ ∈[0, 1]. Proof of Lemma 3.3.Letting z=x+iy, x, y∈R, we have |z|2pz=x2+y2px+ ix2+y2py. For f(x, y) =x2+y2px, define the function g:[0, 1] →Rby g(λ) = f(λx+(1 −λ)x, λy +(1 −λ)y). Then, by the Fundamental Theorem of Calculus, 140
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 x2+y2p x−x2+y2p x≡f(x,y) −f(x,y)≡g(1)−g(0)= 1 0 dg dλ dλ. Furthermore, by the chain rule, for each (x, y), (x, y) ∈R2we have dg dλ =x−x∂f ∂x(λx +(1−λ)x,λy+(1−λ)y) +y−y∂f ∂y (λx +(1−λ)x,λy+(1−λ)y). Thus, computing ∂f ∂x (x, y) =2px2+y2p−1x2+x2+y2p, ∂f ∂y (x, y) =2px2+y2p−1xy and letting Xλ=λx +(1−λ)x, Yλ=λy +(1−λ)yso that Xλ+iYλ=λz +(1 −λ)z=:Zλ, we obtain x2+y2p x−x2+y2p x=⎛ ⎝ 1 02p|Zλ|2(p−1)X2 λ+|Zλ|2pdλ⎞ ⎠x−x +⎛ ⎝ 1 0 2p|Zλ|2(p−1)XλYλdλ⎞ ⎠y−y. In addition, the symmetry in xand yreadily implies x2+y2p y−x2+y2p y=⎛ ⎝ 1 02p|Zλ|2(p−1)Y2 λ+|Zλ|2pdλ⎞ ⎠y−y +⎛ ⎝ 1 0 2p|Zλ|2(p−1)XλYλdλ⎞ ⎠x−x. The last two equations can be combined to yield |z|2pz−|z|2pz= 1 0 |Zλ|2pdλ!z−z+2p 1 0 |Zλ|2(p−1)X2 λx−x+XλYλy−y +iYλy−y+iXλYλx−xdλ, which is the desired expression since, upon completing the square, the quantity inside the square bracket on the right-hand side is equal to 1 2Z2 λ(z−z)+1 2|Zλ|2z−z. Returning to (3.48), we employ Lemma 3.3 with z= +ζand z= +ζ, which imply Zλ=λ( +ζ) +(1 −λ)( +ζ) =λ +(1 −λ) +ζ, to obtain 141
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 ∂x|+ζ|2p(+ζ)−|+ζ|2p(+ζ)L2(R) ≤(p +1)∂x" 1 0 |Zλ|2pdλ!(−)#L2(R) +p∂x" 1 0 |Zλ|2(p−1)Z2 λdλ!(−)#L2(R) ≤(p +1) 1 0∂x|Zλ|2p(−)L2(R)dλ +p 1 0∂x|Zλ|2(p−1)Z2 λ(−)L2(R)dλ ≤(p +1)sup λ∈[0,1]∂x|Zλ|2p(−)L2(R) +psup λ∈[0,1]∂x|Zλ|2(p−1)Z2 λ(−)L2(R). Thus, by the product rule, ∂x|+ζ|2p(+ζ)−|+ζ|2p(+ζ)L2(R) ≤(p +1)sup λ∈[0,1]∂x|Zλ|2p·(−)L2(R)+|Zλ|2p(−)xL2(R) +psup λ∈[0,1]∂x|Zλ|2(p−1)Z2 λ·(−)L2(R)+|Zλ|2(p−1)Z2 λ(−)xL2(R). (3.50) The second and fourth norms on the right-hand side of (3.50)are easy to handle as follows: |Zλ|2p(−)xL2(R)=|Zλ|2(p−1)Z2 λ(−)xL2(R) ≤Zλ2p L∞(R)(−)xL2(R) ≤H1(R)+H1(R)+ζL∞(R)2p−H1(R). (3.51) In order to estimate the first and third norms on the right-hand side of (3.50), we note that ∂x|Zλ|2p=2p|Zλ|2p−1∂x|Zλ|=2p|Zλ|2p−2(∂x|Zλ|)|Zλ|, ∂x|Zλ|2(p−1)=2(p −1)|Zλ|2p−3∂x|Zλ|=2(p −1)|Zλ|2(p−2)(∂x|Zλ|)|Zλ|. 142
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Hence, it suffices to compute (∂x|Zλ|)|Zλ|. For this, we observe that ∂x|Zλ|2=2|Zλ|∂x|Zλ|and so we first compute |Zλ|2=λ||2+(1−λ)2||2+q2 0+λ(1−λ) + +λζ+ζ+(1−λ) ζ +ζ , which upon differentiation yields 2|Zλ|∂x|Zλ|=λ2x+x+λ(1−λ) x+x+λx¯ ζ+xζ +λ(1−λ) x+x+(1−λ)2x+x+(1−λ) x¯ ζ+xζ +λζ+¯ ζ+(1−λ) ζ+¯ ζ+ζ¯ ζ+¯ ζζ.(3.52) Therefore, ∂x|Zλ|2p·(−)L2(R)≤2p−L∞(R)|Zλ|2(p−1)(∂x|Zλ|)|Zλ|L2(R) ≤p−L∞(R)Zλ2(p−1) L∞(R)x+xL2(R)+x+xL2(R) +x¯ ζ+xζL2(R)+x+xL2(R)+x+xL2(R) +x¯ ζ+xζL2(R)+ζ +¯ ζL2(R)+ζ +¯ ζL2(R)+ζ¯ ζ+¯ ζζL2(R) ≤2p−L∞(R)Zλ2(p−1) L∞(R)L∞(R)+L∞(R)+ζL∞(R) ·xL2(R)+xL2(R)+ζL2(R)(3.53) hence, noting that |Zλ| ≤|| +|| +|ζ|and using the embedding H1(R) ⊂L∞(R), we obtain the following bound for the first norm on the right-hand side of (3.50): ∂x|Zλ|2p·(−)L2(R)≤2p−H1(R)H1(R)+H1(R)+ζL∞(R)2p−1 ·H1(R)+H1(R)+ζL2(R).(3.54) Proceeding to the third norm on the right-hand side of (3.50), we have ∂x|Zλ|2(p−1)Z2 λ·(−)L2(R)≤−L∞(R)∂x|Zλ|2(p−1)Z2 λL2(R) ≤−L∞(R)2(p −1)|Zλ|2(p−2)|Zλ|(∂x|Zλ|)Z2 λL2(R) +|Zλ|2(p−1)·2Zλ(∂xZλ)L2(R) ≤−L∞(R)(p −1)Zλ2(p−1) L∞(R)2|Zλ|(∂x|Zλ|)L2(R) 143
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 +2Zλ2p−1 L∞(R)∂xZλL2(R). The first of the L2norms on the right-hand side can be handled along the lines of (3.53). For the second L2norm, we have ∂xZλL2(R)=λx+(1−λ)x+ζL2(R)≤H1(R)+H1(R)+ζL2(R). Thus, using once again the embedding H1(R) ⊂L∞(R), we obtain the bound ∂x|Zλ|2(p−1)Z2 λ·(−)L2(R) ≤2p−H1(R)H1(R)+H1(R)+ζL∞(R)2p−1 ·H1(R)+H1(R)+ζL2(R).(3.55) Overall, the bounds (3.51), (3.54) and (3.55) combine with inequality (3.50)to yield ∂x|+ζ|2p(+ζ)−|+ζ|2p(+ζ)L2(R) ≤(2p+1)2H1(R)+H1(R)+ζL∞(R)+ζL2(R)2p−H1(R) which together with inequality (3.48) implies ∂xGF, −GF,L2(R) ≤(2p+1)2H1(R)+H1(R)+ζL∞(R)+ζL2(R)+q02p−H1(R). (3.56) Hence, in view of the L2estimate (3.38), we have GF, −GF,H1(R)≤22p+1p+(2p+1)2H1(R)+H1(R) +ζL∞(R)+ζL2(R)+q02p·−H1(R). In turn, inequality (3.35) yields [](t) −[](t)H1(R)≤2γ22p+1p+(2p+1)2t ·sup τ∈[0,t](τ)H1(R)+(τ)H1(R) +ζL∞(R)+ζL2(R)+q02p( −)(τ)H1(R), which ensures that is a contraction in the ball B(0, ρ) ⊂C([0, Tf]; H1(R)) provided that 144
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Tf<1 2γ22p+1p+(2p+1)22ρ+ζL∞(R)+ζL2(R)+q02p.(3.57) The proof of the local H1existence Theorem 3.1 is complete. Remark 3.1. The bounds (3.47) and (3.57)for the lifespan Tfget larger as the norms 0H1(R), ζL∞(R), ζL2(R), |ζ| −q0L2(R)and the background q0get smaller. This is consistent with the fact that, when all of these quantities are small, the problem heuristically approximates the one with zero boundary conditions, for which global existence is ensured for all p≥1 and sufficiently small initial data according to the known results of Theorem 2.1. Specifically, when ζ=q0=0 and 1 ≤p<2, the arguments of Theorem 3.1 provide the first step towards establishing global existence of solutions satisfying the size estimate (2.2). In particular, this first step implies local existence of solutions satisfying (2.2). This result is then extended to global existence by employing appropriate conservation laws, which are useful in the context of zero boundary conditions unlike the case of nonzero boundary conditions (see discussion in the next section). Closeness estimates for finite times. Combining the closeness estimates (3.22) and (3.26) with the local existence result of Theorem 3.1, we arrive at the following analogue of Theorem 1.1 for the case of nonzero boundary conditions at infinity. Theorem 3.2 (Theorem 1.1 for the nonzero boundary conditions (3.1)). Consider the modified NLS equations (3.11)and (3.12)with initial data φ0, 0∈H1(R)given by (3.13)and the boundary conditions (3.14), in the case of either the saturable nonlinearity F(x) =x κ(1+x) or the power nonlinearity F(x) =xpwith p≥1. (i) L2closeness: Given 0 <ε<1, suppose that the initial data satisfy φ0−0L2(R)≤Cε3,(3.58) φ0H1(R)≤c0ε, 0H1(R)≤C0ε(3.59) for some constants C, c0, C0>0. In addition, suppose that the nonzero background described by the function ζ∈X1(R)satisfies q0≤Bε, ζL∞(R)≤B0ε, ζL2(R)≤B1ε, |ζ|−q0L2(R)≤B2ε(3.60) for some constants B, B0, B1, B2>0. Then, there exists a finite time Tc∈0,T f, where Tfis the lifespan of solutions to the non-integrable NLS equation (3.12)from Theorem 3.1, and a constant C= C(μ, γ, c0, C0, C, B, B0, B1, B2, Tc)such that the solutions φ(x, t) and (x, t) satisfy the closeness estimate sup t∈[0,Tc]φ(t)−(t)L2(R)≡sup t∈[0,Tc]e−iμq2 0tu(t) −e−iγF(q2 0)t U(t)L2(R)≤ Cε3.(3.61) (ii) H1and L∞closeness: If the initial data φ0, 0satisfy (3.59)along with the stronger condition (in place of (3.58)) 145
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 φ0−0H1(R)≤C1ε3(3.62) for some constant C1>0and, in addition, (3.60)holds, then there exists a constant C1 depending on C1and with a similar dependency on Tcand μ, γ, c0, C0, B0, B1, B2as the constant Cin (3.61)such that sup t∈[0,Tc]φ(t)−(t)H1(R)≡sup t∈[0,Tc]e−iμq2 0tu(t) −e−iγF(q2 0)t U(t)H1(R)≤ C1ε3. (3.63) Consequently, there exists a constant C2with similar dependencies as C1such that sup t∈[0,Tc]φ(t)−(t)L∞(R)≡sup t∈[0,Tc]e−iμq2 0tu(t) −e−iγF(q2 0)t U(t)L∞(R)≤ C2ε3. (3.64) Proof. We only give the details for part (ii), as part (i) can be established similarly. The solutions to the modified cubic NLS equation (3.11)exist globally in time with φ∈C([0, ∞); H1(R)). On the other hand, thanks to Theorem 3.1, the solutions to the modified general NLS equation exist at least locally with ∈C([0, Tf]; H1(R)). Consider both solutions on the time interval [0, Tf]and suppose that the conditions (3.59)are met. Then, since both solutions belong to C([0, Tf]; H1(R)), by continuity there exist T1, T2∈[0, Tf]associated with the solutions φ, of the integrable and non-integrable NLS equations, respectively, such that φ(t)H1(R)≤˜c0ε∀t∈[0,T 1], (t)H1(R)≤˜ C0ε∀t∈[0,T 2],(3.65) for some constants ˜c0, ˜ C0>0 independent of t∈[0, T1]and t∈[0, T2], respectively. Let Tc:= min {T1,T 2}. Then, estimates (3.65) hold for all t∈[0, Tc]and can be combined with the closeness estimate (3.26)to imply (t)H1(R)≤M1Tcε2p+1+M2ε2p+1+M3ε3,(3.66) for some constants Mj(μ, γ, C1, B0, B1, B2), j=1, 2, 3 and all t∈[0, Tc](recall that for the saturable nonlinearity p=1, while for the power nonlinearity p≥1). Remark 3.2. The fact that the NLS solutions belong to C([0, Tf]; H1(R)) is crucial, since it allows us to establish that solutions starting from initial data that satisfy the smallness conditions (3.59) remain small in the sense of the bounds (3.65), at least for short times. Such an argument could not be implemented for solutions that are not continuous with respect to time, e.g. for solutions belonging in a weaker class such as L2([0, Tf]; H1(R)). Moreover, estimates (3.61), (3.63) and (3.64) highlight that the rotations e−iμq2 0tand e−iγF(q2 0)t are necessary in order to establish the closeness between the solutions of the original NLS equations (1.3) and (1.9)or (1.10)in the case of the nonzero boundary conditions (3.1). Indeed, these rotations are the result of converting the non-vanishing conditions (3.1)into vanishing ones via the changes of variables (3.6) and (3.10), which lead to the modified problems 146
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Thus, in view of the inequality x2 L2(I) ≤2 x+ζ2 L2(I) +2 ζ2 L2(I), sup t∈[0,Tmax)x(t)2 L2(I) ≤20x+ζ2 L2(I) +4C402p+2 L2p+2(I) +4C1R02p+1 L4p+2(I) +4C20p+2 L2(I) 0xp L2(I) +4C1|ζ|−q0L2(I)R+0L2(I ) +2ζ2 L2(I) +4C5Rp+2sup t∈[0,Tmax)x(t)p L2(I) .(4.23) Note that the various Lebesgue norms on the right-hand side can be controlled by the L∞(I) norm (and hence by the H1 0(I) norm) due to the crucial fact that we are working on the finite domain I. In the subcritical case 1 ≤p<2, dividing (4.23)by supt∈[0,Tmax)x(t)p L2(I) we obtain sup t∈[0,Tmax)x(t)2−p L2(I) ≤4C5Rp+2+1 supt∈[0,Tmax)x(t)p L2(I) 20x+ζ2 L2(I) +4C402p+2 L2p+2(I) +4C1R02p+1 L4p+2(I) +4C20p+2 L2(I) 0xp L2(I) +4C1|ζ|−q0L2(I)R+0L2(I ) +2ζ2 L2(I) ,(4.24) which shows that supt∈[0,Tmax)x(t)L2(I) <∞and hence implies global existence in H1 0(I). Indeed, if supt∈[0,Tmax)x(t)L2(I) =∞then, due to the fact that 2 −p>0 and p>0, (4.24) leads to the contradiction ∞ ≤4C5Rp+2. Thus, part (i) of the theorem has been proved. In the critical case p=2, (4.23) can be rearranged to yield the uniform bound 1−4C5R4sup t∈[0,Tmax)x(t)2 L2(I) ≤20x+ζ2 L2(I) +4C406 L6(I) +4C1R05 L10(I ) +4C204 L2(I) 0x2 L2(I) +4C1|ζ|−q0L2(I )R+0L2(I) +2ζ2 L2(I) (4.25) and so we once again deduce supt∈[0,Tmax)x(t)L2(I ) <∞, but now provided that 1 − 4C5Rp+2>0, which is precisely the second condition in (4.15) (note that this condition is realizable thanks to the first condition in (4.15)). This implies global well-posedness in H1 0(I) and hence completes the proof of part (ii) of the theorem with C=C5(which depends only on pand ζL∞(I)). We remark that if p>2 then neither of the two arguments used in the proof of Theorem 4.1 works. Theorem 4.1 has the following implication concerning the proximity of solutions to the integrable and non-integrable NLS equations in the case of a finite domain. 153
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Theorem 4.2 (Theorem 1.1 for the finite interval). Given 0 <ε<1, suppose that the hypothesis of Theorem 3.2 with the relevant norms over Rreplaced with their counterparts over the finite interval I=(−L, L) holds true. If L =O1 ε, then the closeness estimates of Theorem 3.2 are valid on Ifor arbitrary 0 <T f<∞. Proof. Since ζL∞(I) =O(ε) and L =O1 εby hypothesis, the second condition in (4.15)is satisfied for sufficiently small initial data such that 0L2(I) =O(ε) and thus we can invoke the global existence result of Theorem 4.1. In particular, we may assume that 0H1 0(I) =O(ε) so that the form of the right-hand side of estimates (4.24) and (4.25) guarantees (at least) (t)H1 0(I) =O(ε). Then, assuming also the condition (3.62) on the H1(I) distance between the initial data of the integrable and non-integrable models, we can derive the closeness estimates exactly as in the proof of Theorem 3.2. Now, however, thanks to the global existence result of Theorem 4.1, we can replace Tcby any arbitrary finite time 0 <T f<∞. The periodic problem. We turn our attention to the periodic Cauchy problem, namely to the case where the integrable and non-integrable NLS equations (1.3) and (1.4)are supplemented with the periodic boundary conditions u(−L,t) =u(L, t), U (−L,t) =U(L,t), t ≥0.(4.26) Importantly, the counterpart of Theorem 2.1 for the periodic conditions (4.26), which guarantees global solutions at the level of the H1norm with the same estimate as (2.2), is proved in Theorem 2.1 of [42]. With this global existence result in hand, the analogue of Theorem 1.1 on the proximity of solutions to the integrable and non-integrable NLS equations in the case of the periodic boundary conditions (4.26) can be stated as follows. Theorem 4.3 (Theorem 1.1 for the periodic Cauchy problem). For p≥1and ν, μ, γ>0, consider the integrable and non-integrable focusing NLS equations (1.3)and (1.4)with x∈I= (−L, L), the initial conditions u(x, 0) =u0(x), U(x, 0) =U0(x), and the periodic boundary conditions (4.26). (i) L2closeness: Let 0 <ε<1and suppose that the initial data satisfy u0−U0L2(I) ≤Cε3,(4.27) u0H1(I) ≤c0ε, U0H1(I) ≤C0ε, (4.28) for some constants c0, C0, C>0. Then, for arbitrary finite 0 <T f<∞, there exists a constant C=C(μ, γ, c0, C0, C, Tf)such that sup t∈[0,Tf]u(t) −U(t)L2(I ) ≤ Cε3.(4.29) (ii) H1and L∞closeness: Suppose that the initial data satisfy (4.28)and the stronger condition (instead of (4.27)) u0−U0H1(I) ≤C1ε3(4.30) 154
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 for some constant C1. Then, there exists a constant C1with similar dependencies as Cin (2.5)such that sup t∈[0,Tf]u(t) −U(t)H1(I ) ≤ C1ε3,(4.31) and, in turn, sup t∈[0,Tf]u(t) −U(t)L∞(I ) ≤ C2ε3.(4.32) Proof. The argument is entirely analogous to the one used for the proof of Theorem 2.2 in the non-periodic case. In particular, expressing periodic functions ϕ∈L2(I) in the form of the Fourier series ϕ(x) =$ ξ∈Z eiπ Lξx ϕ(ξ) where now ϕ(ξ) =1 2L L −L e−iπ Lξx ϕ(x)dx, we find that the difference of solutions =u −Uto the integrable and non-integrable NLS equations satisfies (ξ, t) =e−iνπ Lξ2t (ξ, 0)−i t 0 e−iνπ Lξ2(t−τ) N(ξ,τ)dτ, (4.33) where N(ξ, t) denotes the spatial Fourier series of the nonlinearity function N(x, t) defined by (2.9). Thus, since (t)L2(I) =√2L (t)2(Z)by Parseval’s theorem, using Minkowski’s integral inequality and the fact that eiνπ Lξ2tis unitary, we have (t)L2(I) ≤(0)L2(I) +tsup τ∈[0,t]N(τ)L2(I) .(4.34) Then, similarly to the steps that led to estimate (2.16), we employ (1.5) and the analogue of the embedding (2.15)on Ifor q=6 and also for q=2(p +1)to obtain N(t)2 L2(I) ≤2μ2u(t)6 L6(I) +2γ2K2U(t)2(2p+1) L2(2p+1)(I) ≤2μ2u(t)6 H1(I) +2γ2K2U(t)2(2p+1) H1(I) from which we infer (t)L2(I) ≤(0)L2(I) +Asup τ∈[0,t]u(τ)3 H1(I) +U(τ)2p+1 H1(I)t with A =√2max{μ, γK}as before. Therefore, for initial data u0, U0satisfying (4.27) and (4.28), using the periodic analogue of Theorem 2.1 and the solution size estimate (2.2)(see Theorem 2.1 on page 662 of [42]), we deduce the desired L2bound (4.29), namely (t)L2(I) ≤Cε3+AM3c3 0ε3+M2p+1C2p+1 0ε2p+1t≤ Cε3,t∈[0,T f], 155
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 with C=max C, AM3c3 0Tf, AM2p+1C2p+1 0Tf. In order to prove (4.31), we combine the definition of the H1(I) norm with (4.33)to infer (t)2 H1(I) ≤2$ ξ∈Z1+ξ2 (ξ, 0)2+2$ ξ∈Z1+ξ2 t 0 N(ξ,τ)dτ!2 . Hence, by Minkowski’s integral inequality, (t)2 H1(I) ≤2(0)H1(I) +2t2sup τ∈[0,t]N(τ)H1(I) and proceeding as for (2.16) and (2.21)we eventually obtain the analogue of (2.7). Finally, the L∞estimate (4.32) follows from the H1estimate (4.31)via the Sobolev embedding theorem. Numerical studies for the case of nonzero boundary conditions. We now present the results of numerical studies in the case of nonzero boundary conditions in order to illustrate (i) our theoretical results of Section 3on the nonzero boundary conditions on the whole line and (ii) the preceding theoretical results of the current section concerning the approximations of the problem on the infinite line by finite intervals. The numerical studies are motivated by the numerical investigations of [11] and [9]. We consider the simplest case where ζ(x) =q0>0. We will treat two examples of initial conditions decaying on the nonzero background, namely exponentially and algebraically decaying ones, the latter stemming from the Peregrine soliton. I. Exponentially decaying initial data on the top of the nonzero background q0. We study the dynamics of the NLS equations emerging from the initial condition of the form u(x, 0)=q0(1+isechx) =U(x,0). (4.35) In all the numerical results we set μ =γ=1for the nonlinearity parameters of the integrable and non-integrable NLS equations, and ν=1, for the linear dispersion parameter. For the amplitude of the background we set q0=0.25 in order to comply with the smallness conditions on the initial data of Theorem 3.2. Based on the analysis presented in Section 3and Theorem 3.2, we may expect closeness between the solutions of the integrable and the non-integrable NLS for short times. Thus, we start with the presentation of the numerical results for a short time interval t∈[0, 10]. Fig. 4.1 depicts contour plots of the spatiotemporal evolution of the density of the NLS equations. The left panel depicts the dynamics for the integrable NLS, the central panel for the non-integrable NLS with the subcritical power nonlinearity (1.9) with p=3/2 (quartic case), and the right panel for the non-integrable NLS (1.10) with saturable nonlinearity. The patterns look almost indistinguishable and we expect this to be verified by the evolution of norms of the distance function. This is indeed the case, as it is shown in Fig. 4.2. Due to estimate (3.61)of Theorem 3.2, we examine the evolution of the norm (t)X:=e−iμq2 0tu(t) −e−iγF(q2 0)t U(t)X.(4.36) For all norms we observe an excellent agreement with the predictions of Theorem 3.2 regarding their linear growth. On the other hand, we find again the moderate increase of (t)L∞and the 156
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 4.1. Contour plots of the spatiotemporal evolution of the initial condition (4.35) with q0=0.25. Left: Integrable NLS p=1. Center: Non-integrable NLS (1.9) with power nonlinearity in the subcritical case p=3/2. Right: Non-integrable NLS (1.10) with saturable nonlinearity. Fig. 4.2. Evolution of the distance norm (4.36)for the dynamics shown in Fig. 4.1. Left: (t)L∞. Center: (t)L2. Right: (t)H1. Fig. 4.3. Contour plots of the spatiotemporal evolution of the initial condition (4.35) with q0=0.25 for longer times t∈[0, 400]. Left: Integrable NLS p=1. Center: Non-integrable NLS (1.9) with power nonlinearity in the subcritical case p=3/2. Right: Non-integrable NLS (1.10) with saturable nonlinearity. larger growth of (t)L2and (t)H1. Again the saturable model exhibits dynamics which are closer to the integrable one than the quartic NLS, while both non-integrable systems are closer to the integrable one in the sense of the pointwise topology. The larger, although still mediocre deviation in the L2and H1norms, is a first indication that in the non-integrable dynamics there may be smaller, finer structures or oscillations present in the solutions that are not captured by the large-scale pattern. These smaller scales could be associated with faster oscillations or sharper gradients in the solutions of the non-integrable NLS. The existence of such scales can become more eminent over longer time intervals. To investigate further these issues, we extended considerably the time-horizon of the above numerical investigations. Fig. 4.3 extends the study presented in Fig. 4.1 to a time span t∈[0, 400]. The results of Fig. 4.3 verify that the evolution observed for t∈[0, 10]is the initial stage of the modulational instability dynamics discussed in detail in [11] and [9]. In each panel, the bounding 157
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 4.4. Evolution of the distance norm (4.36)for the dynamics of Fig. 4.3 over t∈[0, 400]. Left: (t)L∞. Center: (t)L2. Right: (t)H1. Fig. 4.5. Evolution of the distance norm (4.36)for the dynamics of Fig. 4.3 over t∈[0, 400], but with the norm evaluated over a spatial interval of length O(1/ε), namely x∈(−1/ε, 1/ε) =(−4, 4). Left: (t)L∞(−4,4). Center: (t)L2(−4,4). Right: (t)H1(−4,4). Fig. 4.6. Contour plots of the spatiotemporal evolution of the initial condition (4.39) with δ=ν=1, γ=μ =1, q0=0.3 and T0=−25 for t∈[−50, 100]. Left: Integrable NLS p=1. Center: Non-integrable NLS (1.9) with power nonlinearity in the subcritical case p=3/2. Right: Non-integrable NLS (1.10) with saturable nonlinearity. lines represent the linear caustics separating the xt-plane into two types of regions: a left far-field region and a right far-field region, where the solution meets the condition at infinity (up to a phase shift), and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. Although at the level of the nonlinear stage of modulational instability the non-integrable dynamics shares to a great extent the basic features of the ones of the integrable counterpart, there are still some important findings revealed. Comparing the pattern of the integrable NLS (shown in the left panel) and the corresponding one for the saturable nonlinearity (right panel), we observe that they have more in common than the pattern for the quartic nonlinearity and the integrable one (central panel). The long-time behavior of the saturable model confirms that, in the case of nonzero boundary conditions too, this is the more structurally stable model in reference to the integrable one when compared to its power nonlinearity counterpart. For the quartic NLS, the oscillatory pattern within the caustics region is markedly different as it 158
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 4.7. Snapshots of the evolution of the initial condition (4.39) with δ=ν=1, γ=μ =1, q0=0.3and T0=−25, portraying the comparison between the profiles of the solutions of the integrable NLS with p=1, the non-integrable NLS with power nonlinearity p=3/2, and the non-integrable NLS with saturable nonlinearity. The bottom right panel depicts the comparison of the profiles of the first event in each of the panels of Fig. 4.6, which occurs at ˜ t=25 for the integrable case p=1, ˜ t∼30 for the power nonlinearity p=3/2, and ˜ t∼25 for the saturable case. exhibits a considerably longer period of the internal modulated wave (i.e. longer times between the peak events) than in the other systems. On the other hand, the maximum amplitude of the internal oscillations seems to be similar in all systems indicating that the similarity of all patterns at large amplitudes should be manifested in the L∞closeness while the differences of the patterns in their finer structures should be reflected in more pronounced deviations in the L2and H1norms. In connection with these predictions, we illustrate in Fig. 4.4 the evolution of the distance (t)Xassociated with the dynamics shown in Fig. 4.3. From the central panel, showing the evolution of (t)L∞it is evident that the non-integrable modulational instability dynamics are remarkably close to the integrable ones in the sense of the pointwise topology, as (t)L∞ initially grows linearly but then becomes bounded and at most of order ∼O(10−1). Concerning (t)L2(central panel) and (t)H1(right panel), we observe an almost linear growth for t∈[0, 100]and a large deviation in compliance with the differences of the oscillatory behavior seen in Fig. 4.3. Revisiting the latter in the saturable and integrable cases, there are important differences between their oscillations close to the bounding caustics. These differences explain the deviation of the systems in the L2and H1topology. On the other hand, the patterns of the 159
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 4.8. Contour plots of the spatiotemporal evolution of the initial condition (4.39) with δ=2, ν=1, γ=μ =1, q0=0.25 and T0=50 for t∈[−50, 200]. Left: Integrable NLS p=1. Center: Non-integrable NLS (1.9) with power nonlinearity in the subcritical case p=3/2. Right: Non-integrable NLS (1.10) with saturable nonlinearity. Fig. 4.9. Evolution of the distance norm (t)Xdefined in (4.36)for the dynamics of the initial condition (4.39) with the parameters of Fig. 4.8, x∈[−4, 4]and t∈[−50, 250]. Left: (t)L∞. Center: (t)L2. Right: (t)H1. integrable and the saturable NLS look very similar in the core, close to x=0. This observation suggests an agreement with the theoretical claim of Theorem 4.1 for the bounded domain approximation, that the dynamics should be more similar when considered on finite intervals of length L ∼O1 ε(cf. Theorem 4.2). The fact that, when restricted to an interval of length ∼O1 ε, the similarity is significantly more noticeable than on the full length interval L, is confirmed in Fig. 4.5. With our choice of ζL∞=q0=0.25 we monitor the behavior of the distances in the restricted interval for x∈[−4, 4]in accordance with Theorems 4.1 and 4.2. The plots feature the remarkable decrease of the deviation from the integrable dynamics particularly for the (t)L2 and (t)H1norms; in contrast with the plots of Fig. 4.4, the deviation in these norms seems is bounded and at most of order O(1), suggesting that on large spatial scales, most of the deviation stems from the finer oscillating structures far from the core of the pattern. II. Algebraically decaying initial data on top of the nonzero background q0.We conclude with the presentation of a numerical study concerning the dynamics emerging from a quadratically decaying initial condition on top of the finite background q0>0. For this purpose, we will use initial conditions defined by the Peregrine soliton solution-the famous rational solution of the integrable NLS iut+δ 2uxx +μ|u|2u=0,(4.37) given by the formula 160
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 Fig. 4.10. Simulation of the spatiotemporal evolution of the density of the solutions to the integrable and non-integrable NLS equations with the nonzero boundary conditions (3.1) over the non-constant background ζ(x) of the form (4.40) and the initial conditions (4.41) (corresponding to the modified NLS equations (3.11)and (3.12)). Parameters A1=0.3, β=0.01, q0=0.1, A2=0.1. Top left panel: Integrable (cubic) NLS. Top right panel: Quartic NLS (p=3/2). Bottom left panel: Saturable NLS. Bottom right panel: Evolution of the L∞norm of the distance function for x∈[−200, 200] and t∈[0, 400]. More details in the text. uPS(x, t;δ;μ;T0;q0)=q0%1−41+2i(t+T0) 1+4x2 K2 0+4(t+T0)2 2&e i(t+T0) .(4.38) The parameters in (4.38)are =1 μq2 0 , K0=√δ, and T0is a time-translation. Comparing the dispersion coefficients of the NLS equations (1.9) and (4.37), we see that they coincide when δ=2ν. With this observation we distinguish between two examples. (a) δ=2ν. We consider the parameters δ=ν=1, γ=μ =1, q0=0.3 and T0=−25, in order to be compliant with the smallness conditions on the initial data when considering the initial condition u(x, 0)=uPS(x, 0;δ;μ;T0;q0), (4.39) for the above set of parameters. This is an example where (4.38)is not an analytical solution of the NLS (1.9)for its integrable case p=1. The theoretical results of [11] and [9] establish very similar spatiotemporal behavior of the integrable NLS as in the case I, and this fact is confirmed in the left panel of Fig. 4.6. Comparing 161
D. Hennig, N.I. Karachalios, D. Mantzavinos et al. Journal of Differential Equations 397 (2024) 106–165 this panel with the central one which corresponds to the case p=3/2 and the right one which corresponds to the saturable NLS, we again observe that the saturable dynamics is closer to the integrable one than its power nonlinearity counterpart. The first events of large amplitude occurring in all contour plots suggest also the presence of features reminiscent of Peregrine rogue waves (PRWs) governed by the specific type of the initial condition. To highlight this feature, as well as the similarities and differences of the solutions, we present in Fig. 4.7 snapshots of the profiles of the solutions for specific times. The similarities to PRWs can be observed for times around t=25 where the function (4.38) attains its maximum as shown in the fourth panel (t=˜ t∼30 for the quartic and t=˜ t∼25 for the saturable). For larger times (t=50, third panel), the differences of the profiles are reflected in the larger deviation of the norms of the distance function between the solutions. The proximity around the center x0=0 predicted by Theorem 4.2 is also illustrated by the snapshots exhibiting the similarity of the central spikes. (b) δ=2ν. For μ =γ=1, we choose δ=2, q0=0.25 and T0=50 in the formula (4.38) and ν=1for the NLS (1.9). Now, the formula (4.38) defines an analytical solution of the NLS (1.9) for its integrable case p=1. Using the initial condition (4.39)for this choice of parameters we observe crucial differences in comparison with the dynamics of the case (a), as it can be seen in Fig. 4.8, for long times. In the left panel, which shows the dynamics for the integrable case p=1, the PRW attains it maximum at t=0, since we have triggered initially, the exact analytical PRW solution and the MI effects predicted by [11,9] should occur much later. This is not the case for the non-integrable models whose dynamics are represented in the central panel (power nonlinearity with p=3/2) and the right panel (saturable), respectively. Fig. 4.9, showing the time evolution of the norms of the distances for x∈[−4, 4], further confirms for the initial condition (4.39), that when considering sufficiently small spatial scales and small time intervals, one observes diminished deviations between non-integrable and integrable dynamics in all of the distances (t)X. Particularly, for t∈[−50, 50], the behavior of the norms for the saturable NLS suggests the proximity of the first RW occurring close to t=0, as in the integrable case p=1. For the quartic nonlinearity, the first RW appears much later at t∼50, in compliance with the larger deviation of norms. However, both patterns and norms show that for t∈[−50, 50], the dynamics of all systems are similar. This is an illustration that Theorems 3.2 and 4.2 may capture important nonlinear effects, particularly when these occur within short time scales. III. Dynamics for non-constant ζ(x). We conclude with a brief presentation of numerical results concerning the case of the nonzero boundary conditions (4.1), now with a non-constant ζ(x). We consider a simple example of a non-constant background ζ(x)=A1exp(−βx2)+q0,A 1,β,q 0>0,(4.40) so that ζ(x) →q0>0at an exponential rate as |x| →∞. For ζ(x) given by (4.40), we solve numerically the finite interval problems approximating the ones on the infinite line for the modified NLS equations (3.11) and (3.12) supplemented with the nonzero boundary conditions (4.1). The corresponding initial conditions are φ(x,0)=(x, 0)=iA2sechx, A2>0,(4.41) which approximately satisfy the zero Dirichlet boundary conditions (4.3)for large L. We return to the solutions uand Uof the original equations (1.3) and (1.4) with the nonzero boundary conditions (3.1)via the transformations (3.6) and (3.10). The top left panel of Fig. 4.10 shows the 162