Physica D: Nonlinear Phenomena最新文献

{"title":"Long-time asymptotics of the coupled nonlinear Schrödinger equation in a weighted Sobolev space","authors":"Yubin Huang ,&nbsp;Liming Ling ,&nbsp;Xiaoen Zhang","doi":"10.1016/j.physd.2026.135138","DOIUrl":"10.1016/j.physd.2026.135138","url":null,"abstract":"<div><div>We study the Cauchy problem for the focusing coupled nonlinear Schrödinger (CNLS) equation with initial data <strong>q</strong>₀ lying in the weighted Sobolev space and the scattering data having <em>n</em> simple zeros. Based on the corresponding 3 × 3 matrix spectral problem, we deduce the Riemann-Hilbert problem (RHP) for CNLS equation through inverse scattering transform. We remove discrete spectra of initial RHP using Darboux transformations. By applying the nonlinear steepest-descent method for RHP introduced by Deift and Zhou, we compute the long-time asymptotic expansion of the solution <strong>q</strong>(<em>x, t</em>) to an (optimal) residual error of order <span><math><mrow><mi>O</mi><mrow><mo>(</mo><msup><mi>t</mi><mrow><mo>−</mo><mn>3</mn><mo>/</mo><mn>4</mn><mo>+</mo><mn>1</mn><mo>/</mo><mo>(</mo><mn>2</mn><mi>p</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></mrow></math></span> where 2 ≤ <em>p</em> &lt; ∞. The leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions. Our work strengthens and extends the earlier work regarding long-time asymptotics for solutions of the nonlinear Schrödinger equation with a delta potential and even initial data by Deift and Park.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135138"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A data-driven integrable BFGS algorithm (IBA-PDE) for discovering PDEs","authors":"Shifang Tian,&nbsp;Biao Li","doi":"10.1016/j.physd.2026.135135","DOIUrl":"10.1016/j.physd.2026.135135","url":null,"abstract":"<div><div>Data-driven discovery of partial difference equations (PDEs) has become a hot topic, and scholars have proposed some excellent data-driven methods (PINNs,PDE-FIND,DLGA-PDE,SGA-PDE) and achieved good results in discovering PDEs. This paper proposes a new integrable BFGS algorithm (IBA-PDE) for PDE discovery, which solves two key problems: (1) To manage the complexity and redundancy of candidate PDE terms, it incorporates a weight balance condition tailored for partially integrable PDEs, along with a preliminary optimization strategy, we first solve the problem of narrowing down the range of PDEs candidates; (2) To accurately estimate unknown PDEs coefficients, the method employs the BFGS optimization algorithm, enhancing the precision of the identification process. Through systematic numerical experiments, IBA-PDE demonstrates superior capability that not only rediscovers fundamental PDEs but also resolves previously intractable systems with unprecedented precision. Specifically, IBA-PDE discovered several complex integrable PDEs (fifth-order KdV, Kaup Kupershmidt, Sawada Kotera, complex modified KdV, Hirota, and (2+1) dimensional Kadomtsev Petviashvili (KP) equations) and two non integrable PDEs (Burgers KdV and Chafee Infante equations), all of which have mean square errors (MSEs) of <span><math><msup><mn>10</mn><mrow><mo>−</mo><mn>9</mn></mrow></msup></math></span> and coefficient errors of almost zero. Moreover, IBA-PDE use fewer experimental data compared to other data-driven methods throughout the entire process of discovering complete PDEs, whether in the stage of determining PDEs candidate terms or coefficient determination. For non-integrable systems, IBA-PDE employs an adaptive discovery mechanism that not only successfully resolves the Burgers-KdV equation but also autonomously identifies a new PDE that better matches the data of the Chafee-Infante equation reducing MSE from <span><math><msup><mn>10</mn><mrow><mo>−</mo><mn>11</mn></mrow></msup></math></span> to <span><math><msup><mn>10</mn><mrow><mo>−</mo><mn>14</mn></mrow></msup></math></span>. Robustness analysis confirms the method’s stability under noise conditions of 1 %, 3 % and 5 %, maintaining the same MSE levels. IBA-PDE establishes a new paradigm for data-driven PDEs discovery, with transformative potential for discovering new PDEs or matching known PDEs from experimental data in fields such as physics, engineering, mechanics, chemistry and biology.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135135"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Matrix integrable hierarchies connected with the symplectic Lie algebras sp(2m) and their bi-Hamiltonian structures and Darboux transformations","authors":"Wen-Xiu Ma","doi":"10.1016/j.physd.2026.135142","DOIUrl":"10.1016/j.physd.2026.135142","url":null,"abstract":"<div><div>This study introduces a framework of matrix spectral problems associated with the general symplectic Lie algebras sp(2<em>m</em>), and establishes their corresponding integrable hierarchies through the zero-curvature formulation. The trace identity is employed to establish the bi-Hamiltonian structures, while the associated Lax pairs ensure the existence of Darboux transformations. Furthermore, the <em>N</em>-fold Darboux transformation is systematically formulated through iterations of first-order Darboux transformations, and an explicit single-step application is also presented.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135142"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Orbital stability of smooth solitary waves for the modified Camassa-Holm equation","authors":"Xijun Deng ,&nbsp;Stéphane Lafortune ,&nbsp;Zhisu Liu","doi":"10.1016/j.physd.2026.135140","DOIUrl":"10.1016/j.physd.2026.135140","url":null,"abstract":"<div><div>In this paper, we explore the orbital stability of smooth solitary wave solutions to the modified Camassa-Holm equation with cubic nonlinearity. These solutions, which exist on a nonzero constant background <em>k</em>, are unique up to translation for each permissible value of <em>k</em> and wave speed. By leveraging the Hamiltonian nature of the modified Camassa-Holm equation and employing three conserved functionals-comprising an energy and two Casimirs, we establish orbital stability through an analysis of the Vakhitov-Kolokolov condition. This stability pertains to perturbations of the momentum variable in <span><math><mrow><msup><mi>H</mi><mn>1</mn></msup><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135140"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Experiments and simulations on the Richtmyer-Meshkov instability with a thin intermediate layer","authors":"K. Ferguson ,&nbsp;B.J. Colombi ,&nbsp;K.M. Church ,&nbsp;O.B. Shende ,&nbsp;Y. Zhou ,&nbsp;J.W. Jacobs","doi":"10.1016/j.physd.2026.135134","DOIUrl":"10.1016/j.physd.2026.135134","url":null,"abstract":"<div><div>Experiments and simulations of the Richtmyer-Meshkov instability (RMI) in two- and three-layer configurations are presented. The two-layer case utilizes a light-over-heavy configuration and consists of air as the light gas and sulfur hexafluoride (SF₆) as the heavy gas. The three-layer case utilizes a light-intermediate-heavy configuration, with helium (He) as the light gas, air as the intermediate gas, and SF₆ as the heavy gas. Statistically significant differences in the mixing layer width of the lower interface are not observed between the two cases. This differs from the experiments of Schalles et al. [1], where a small, though statistically significant, difference in mixing layer growth was observed between the two- and three-layer cases with a nominally two-dimensional, single mode perturbation. Notably, the perturbations on the lower interface in the present work do not grow large enough to significantly interact with the upper interface during the duration of the experiments. This suggests that the differences in mixing layer growth observed by Schalles et al. <span><span>[1]</span></span> may be due to interactions of the perturbations on one interface with the other interface rather than being inherent to the three-layer problem.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135134"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Estimating the epidemic peak in an infection-age-structured SIR model","authors":"Ali Moussaoui ,&nbsp;Mohammed Mesk ,&nbsp;Shigui Ruan","doi":"10.1016/j.physd.2026.135147","DOIUrl":"10.1016/j.physd.2026.135147","url":null,"abstract":"<div><div>We investigate a susceptitble-infectious-recovered (SIR) epidemic model in which the infectious class is structured by the age of infection. We first establish the final size equation of the epidemic and prove that it admits a unique solution. We then derive two-sided estimates for the epidemic peak and provide lower and upper bounds for the peak time, which behave predictably for large population sizes. In the particular case of constant transmission and recovery rates, these bounds coincide, yielding an explicit expression for the peak time. Our results extend and refine previous findings for the classical SIR model, providing new analytical insights into epidemic dynamics with infection-age structure, which are relevant for mathematical modeling and applied studies in epidemiology.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135147"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The field-dependent wave length of ferrofluidic interfacial instability in magnetic field with diverse gradients","authors":"Liu Li ,&nbsp;Decai Li ,&nbsp;Yunqi Guo","doi":"10.1016/j.physd.2026.135149","DOIUrl":"10.1016/j.physd.2026.135149","url":null,"abstract":"<div><div>The interfacial instability in gradient magnetic fields under gravitational influence has been systematically investigated using finite element analysis. Comparative analysis of simulation and experimental data reveals the coupled effects of gravitational and magnetic forces on instability wave length. The simulation model successfully captures the force balance among magnetic forces, surface tension, and gravity by integrating perturbation field theory with Rosensweig instability framework. Results demonstrate that the interfacial instability wave length in ferrofluid-air systems is jointly determined by magnetic and gravitational forces, as validated through both simulation and experimentation. The findings provide a more universal prediction framework for instability wave lengths across varying magnetic and gravitational field conditions compared to existing theories.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135149"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamics of closed rogue patterns in the Davey-Stewartson I equation","authors":"Weisheng Kong,&nbsp;Lijuan Guo","doi":"10.1016/j.physd.2026.135125","DOIUrl":"10.1016/j.physd.2026.135125","url":null,"abstract":"<div><div>In this paper the formation of the closed rogue patterns in the Davey-Stewartson I equation is investigated. Only one part of these wave structures in the closed rogue waves rises from the constant background and then retreats back to it, and this transient wave possesses patterns such as one ring, doubled ring, one ground and their superposition. But the other part of the wave structure comes from the far distance as some localized lumps, which moves to the near field and interacts with the closed curved waves, and then travels to the large distance again. The closed rogue patterns are determined by the roots of a <em>special</em> polynomial, and the number of lumps at large time could be illustrated by Young diagram. The exact and approximate results show excellent agreement. In addition, we propose that a sufficient and necessary condition to the existence of the closed rogue pattern, namely, it requires <span><math><mrow><msub><mi>core</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>=</mo><mi>⌀</mi></mrow></math></span> and the positive definiteness of a generalized Hermite polynomial.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135125"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the proximal dynamics between integrable and non-integrable members of a generalized Korteweg-de Vries family of equations","authors":"Nikos I. Karachalios ,&nbsp;Dionyssios Mantzavinos ,&nbsp;Jeffrey Oregero","doi":"10.1016/j.physd.2026.135123","DOIUrl":"10.1016/j.physd.2026.135123","url":null,"abstract":"<div><div>The distance between the solutions to the integrable Korteweg-de Vries (KdV) equation and a broad class of non-integrable generalized KdV (gKdV) equations is estimated in appropriate Sobolev spaces. This family of equations includes, as special cases, the standard gKdV equation with power nonlinearities as well as weakly nonlinear perturbations of the KdV equation. For initial data and nonlinearity parameters of arbitrary size, we establish distance estimates based on a crucial size estimate for local gKdV solutions that grows linearly with the norm of the initial data. Consequently, these estimates predict that the dynamics of the gKdV and KdV equations remain close over long time intervals for initial amplitudes approaching unity, while providing an explicit rate of deviation for larger amplitudes. These theoretical results are supported by numerical simulations of one-soliton and two-soliton initial conditions, which show excellent agreement with the theoretical predictions. Furthermore, it is demonstrated that in the case of power nonlinearities and large solitonic initial data, the deviation between the integrable and non-integrable dynamics can be drastically reduced by incorporating suitable rotation effects via a rescaled KdV equation. As a result, the integrable dynamics stemming from the rescaled KdV equation may persist within the gKdV family of equations over remarkably long timescales.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135123"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146081221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multi-front dynamics in spatially inhomogeneous Allen-Cahn equations","authors":"Robbin Bastiaansen ,&nbsp;Arjen Doelman ,&nbsp;Tasso J. Kaper","doi":"10.1016/j.physd.2026.135139","DOIUrl":"10.1016/j.physd.2026.135139","url":null,"abstract":"<div><div>Recent studies of biological, chemical, and physical pattern-forming systems have started to go beyond the classic ‘near onset’ and ‘far from equilibrium’ theories for homogeneous systems to include the effects of spatial heterogeneities. In this article, we build a conceptual understanding of the impact of spatial heterogeneities on the pattern dynamics of reaction-diffusion models. We consider the simplest setting of an explicit, scalar, bi-stable Allen-Cahn equation driven by a general small-amplitude spatially-heterogeneous term ε<em>F</em>(<em>U, U_x, x</em>). In the first part, we perform an analysis of the existence and stability of stationary one-, two-, and <em>N</em>-front patterns for general spatial heterogeneity <em>F</em>(<em>U, U_x, x</em>). Then, for general dynamically-evolving <em>N</em>-front patterns, we explicitly determine the <em>N</em>-th order system of ODEs that governs to leading order the evolution of the front positions. In the second part, we focus on a particular class of spatial heterogeneities where <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>U</mi><mo>,</mo><msub><mi>U</mi><mi>x</mi></msub><mo>,</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>H</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msub><mi>U</mi><mi>x</mi></msub><mo>+</mo><msup><mi>H</mi><mrow><mo>″</mo></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>U</mi></mrow></math></span>, in which <em>H</em> is either spatially localized or spatially periodic. For localized heterogeneities, we determine all stationary <em>N</em>-front patterns, and show that these are unstable for <em>N</em> &gt; 1. We find instead slowly evolving ‘trains’ of <em>N</em>-fronts that collectively travel to  ± ∞, either with slowly decreasing or increasing speeds. For spatially periodic heterogeneities, we show that the fronts of a multi-front pattern will get ‘pinned’ if the distances between successive fronts are sufficiently large, <em>i.e.</em>, the multi-front pattern is attracted to a nearby stable stationary multi-front pattern.</div></div>","PeriodicalId":20050,"journal":{"name":"Physica D: Nonlinear Phenomena","volume":"489 ","pages":"Article 135139"},"PeriodicalIF":2.9,"publicationDate":"2026-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146190293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}