Physica D: Nonlinear Phenomena最新文献

英文中文

A data-driven integrable BFGS algorithm (IBA-PDE) for discovering PDEs 一种数据驱动的可积BFGS算法（IBA-PDE）

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-01-28 DOI: 10.1016/j.physd.2026.135135

Shifang Tian, Biao Li

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

10^{- 9}

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

10^{- 11}

10^{- 14}

. 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.

偏差分方程的数据驱动发现已成为一个热门话题，学者们提出了一些优秀的数据驱动方法（PINNs、PDE-FIND、DLGA-PDE、SGA-PDE），并在偏差分方程的发现方面取得了良好的效果。本文提出了一种新的可积BFGS算法（IBA-PDE），该算法解决了两个关键问题：(1)为管理候选PDE项的复杂性和冗余性，引入了针对部分可积PDE的权重平衡条件，并结合初步的优化策略，首先解决了PDE候选项范围的缩小问题；(2)为了准确估计未知偏微分方程系数，该方法采用BFGS优化算法，提高了识别过程的精度。通过系统的数值实验，IBA-PDE不仅能够重新发现基本的pde，而且能够以前所未有的精度解决以前难以解决的系统。具体来说，IBA-PDE发现了几个复可积偏微分方程（五阶KdV， Kaup Kupershmidt, Sawada Kotera，复修正KdV， Hirota和（2+1）维Kadomtsev Petviashvili （KP）方程）和两个非可积偏微分方程（Burgers KdV和Chafee Infante方程），它们的均方误差（mse）为10−9，系数误差几乎为零。此外，与其他数据驱动的方法相比，IBA-PDE在发现完整pde的整个过程中使用的实验数据更少，无论是在确定pde候选项阶段还是在确定系数阶段。对于非可积分系统，IBA-PDE采用自适应发现机制，不仅可以成功地求解Burgers-KdV方程，还可以自动识别与Chafee-Infante方程数据更匹配的新PDE，将MSE从10−11降低到10−14。鲁棒性分析证实了该方法在1%、3%和5%噪声条件下的稳定性，并保持了相同的MSE水平。IBA-PDE为数据驱动的pde发现建立了一个新的范例，在物理、工程、力学、化学和生物学等领域的实验数据中发现新的pde或匹配已知的pde具有革命性的潜力。

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引用次数: 0

A generalized two-component Novikov system and its analytical properties 广义双分量Novikov系统及其解析性质

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-01-17 DOI: 10.1016/j.physd.2026.135120

Yonghui Zhou , Xiaowan Li , Shuguan Ji , Zhijun Qiao

In this paper, we study the Cauchy problem for a generalized two-component Novikov system with weak dissipation. We first establish the local well-posedness of solutions by using the Kato’s theorem. Then we give the necessary and sufficient condition for the occurrence of wave breaking in a finite time. Finally, we investigate the persistence properties of strong solutions in the weighted

L^{p} (R)

spaces for a large class of moderate weights.

研究一类具有弱耗散的广义双分量Novikov系统的Cauchy问题。首先利用加藤定理建立了解的局部适定性。然后给出了在有限时间内发生破波的充分必要条件。最后，我们研究了一类大的中等权值的加权Lp(R)空间中强解的持久性。

引用次数: 0

Matrix integrable hierarchies connected with the symplectic Lie algebras sp(2m) and their bi-Hamiltonian structures and Darboux transformations 与辛李代数sp（2m）相连的矩阵可积层次及其双哈密顿结构和达布变换

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-02-06 DOI: 10.1016/j.physd.2026.135142

Wen-Xiu Ma

This study introduces a framework of matrix spectral problems associated with the general symplectic Lie algebras sp(2m), 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 N-fold Darboux transformation is systematically formulated through iterations of first-order Darboux transformations, and an explicit single-step application is also presented.

本文引入了一般辛李代数sp（2m）的矩阵谱问题的框架，并通过零曲率公式建立了它们对应的可积层次。利用迹恒等式来建立双哈密顿结构，而相关的Lax对保证了达布变换的存在性。通过一阶Darboux变换的迭代，系统地推导了n次Darboux变换，并给出了一种明确的单步应用。

引用次数: 0

Orbital stability of smooth solitary waves for the modified Camassa-Holm equation 修正Camassa-Holm方程光滑孤立波的轨道稳定性

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-02-05 DOI: 10.1016/j.physd.2026.135140

Xijun Deng , Stéphane Lafortune , Zhisu Liu

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 k, are unique up to translation for each permissible value of k 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

H^{1} (R)

本文研究了具有三次非线性的修正Camassa-Holm方程光滑孤立波解的轨道稳定性。这些解存在于一个非零的恒定背景k上，对于k和波速的每一个允许值都是唯一的。通过利用改进Camassa-Holm方程的哈密顿性质，并采用三个守恒泛函——包括一个能量和两个卡西米尔，我们通过分析Vakhitov-Kolokolov条件建立了轨道稳定性。这种稳定性与H1(R)中动量变量的扰动有关。

引用次数: 0

Experiments and simulations on the Richtmyer-Meshkov instability with a thin intermediate layer 薄中间层richmyer - meshkov不稳定性的实验与模拟

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-01-28 DOI: 10.1016/j.physd.2026.135134

K. Ferguson , B.J. Colombi , K.M. Church , O.B. Shende , Y. Zhou , J.W. Jacobs

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. [1] may be due to interactions of the perturbations on one interface with the other interface rather than being inherent to the three-layer problem.

给出了两层和三层结构中richhtmyer - meshkov不稳定性（RMI）的实验和模拟。双层壳体采用轻过重的结构，由空气作为轻气体和六氟化硫（SF6）作为重气体组成。三层壳体采用轻、中、重结构，以氦（He）为轻气体，空气为中间气体，SF6为重气体。在两种情况下，下界面的混合层宽度没有统计学上的显著差异。这与Schalles等人的实验不同，在他们的实验中，在名义上二维单模扰动的情况下，观察到两层和三层混合层生长的微小差异，尽管统计上显着。值得注意的是，在本工作中，下界面上的扰动没有增长到足以在实验期间与上界面显著相互作用。这表明，Schalles等人观察到的混合层生长的差异可能是由于一个界面上的扰动与另一个界面的相互作用，而不是三层问题固有的。

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引用次数: 0

Estimating the epidemic peak in an infection-age-structured SIR model 在感染年龄结构SIR模型中估计流行高峰

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-02-11 DOI: 10.1016/j.physd.2026.135147

Ali Moussaoui , Mohammed Mesk , Shigui Ruan

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.

我们研究了一个易感-感染-恢复（SIR）流行病模型，其中感染类别是由感染年龄构成的。我们首先建立了流行病的最终大小方程，并证明它有唯一解。然后，我们推导了流行病峰值的双边估计，并提供了峰值时间的下限和上限，这对于大人口规模来说是可预测的。在传输速率和恢复速率恒定的特殊情况下，这些界限重合，产生峰值时间的显式表达式。我们的研究结果扩展和完善了经典SIR模型的先前发现，为感染年龄结构的流行动力学提供了新的分析见解，这与流行病学的数学建模和应用研究相关。

引用次数: 0

The field-dependent wave length of ferrofluidic interfacial instability in magnetic field with diverse gradients 不同梯度磁场下铁磁流体界面不稳定性的场相关波长

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-02-10 DOI: 10.1016/j.physd.2026.135149

Liu Li , Decai Li , Yunqi Guo

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.

用有限元方法系统地研究了重力作用下梯度磁场中界面的不稳定性。仿真和实验数据的对比分析揭示了重力和磁力对不稳定波长的耦合影响。该仿真模型通过将微扰场理论与Rosensweig不稳定性框架相结合，成功地捕获了磁力、表面张力和重力之间的力平衡。结果表明，铁磁流体-空气系统的界面不稳定波长是由磁力和重力共同决定的，并通过仿真和实验验证了这一点。与现有理论相比，这些发现为不同磁场和引力场条件下的不稳定波长提供了一个更普遍的预测框架。

引用次数: 0

Dynamics of closed rogue patterns in the Davey-Stewartson I equation Davey-Stewartson I方程中闭合不规则模式的动力学

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-01-24 DOI: 10.1016/j.physd.2026.135125

Weisheng Kong, Lijuan Guo

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 special 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

{core}_{2} (λ) = ⌀

and the positive definiteness of a generalized Hermite polynomial.

本文研究了Davey-Stewartson 1方程中闭合不规则模式的形成。在封闭的异常波中，这些波结构中只有一部分从恒定背景上升，然后回落，这种瞬态波具有一环、双环、一地及其叠加等模式。但波结构的另一部分来自远场，作为一些局部块，它们移动到近场并与闭合弯曲波相互作用，然后再次传播到大距离。封闭的流氓模式由一个特殊多项式的根决定，大时间内的团块数量可以用Young图表示。精确和近似结果非常吻合。此外，我们还提出了闭流浪模式存在的一个充要条件，即core2(λ)= φ和广义Hermite多项式的正定性。

引用次数: 0

On the proximal dynamics between integrable and non-integrable members of a generalized Korteweg-de Vries family of equations 广义Korteweg-de Vries族方程的可积元与不可积元之间的近端动力学

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-01-21 DOI: 10.1016/j.physd.2026.135123

Nikos I. Karachalios , Dionyssios Mantzavinos , Jeffrey Oregero

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.

在适当的Sobolev空间中估计了可积Korteweg-de Vries （KdV）方程解与广义不可积KdV （gKdV）方程解之间的距离。作为特例，这类方程包括具有幂非线性的标准gKdV方程以及KdV方程的弱非线性扰动。对于初始数据和任意大小的非线性参数，我们建立了基于关键大小估计的局部gKdV解的距离估计，该解随初始数据的范数线性增长。因此，这些估计预测了gKdV和KdV方程的动力学在接近单位的初始振幅的长时间间隔内保持接近，同时为较大的振幅提供了明确的偏差率。这些理论结果得到了单孤子和双孤子初始条件的数值模拟的支持，与理论预测非常吻合。此外，还证明了在幂非线性和大孤子初始数据的情况下，通过重新标度的KdV方程加入适当的旋转效应可以大大减少可积和不可积动力学之间的偏差。结果，源于重标KdV方程的可积动力学可以在gKdV方程族中持续存在很长时间尺度。

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引用次数: 0

Multi-front dynamics in spatially inhomogeneous Allen-Cahn equations 空间非齐次Allen-Cahn方程的多锋面动力学

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-05-01 Epub Date: 2026-02-04 DOI: 10.1016/j.physd.2026.135139

Robbin Bastiaansen , Arjen Doelman , Tasso J. Kaper

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 εF(U, U_x, x). In the first part, we perform an analysis of the existence and stability of stationary one-, two-, and N-front patterns for general spatial heterogeneity F(U, U_x, x). Then, for general dynamically-evolving N-front patterns, we explicitly determine the N-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

F (U, U_{x}, x) = H^{'} (x) U_{x} + H^{″} (x) U

, in which H is either spatially localized or spatially periodic. For localized heterogeneities, we determine all stationary N-front patterns, and show that these are unstable for N > 1. We find instead slowly evolving ‘trains’ of N-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, i.e., the multi-front pattern is attracted to a nearby stable stationary multi-front pattern.

最近对生物、化学和物理模式形成系统的研究已经开始超越均匀系统的经典“近起点”和“远离平衡”理论，包括空间异质性的影响。在本文中，我们从概念上理解了空间异质性对反应扩散模式动力学的影响。我们考虑由一般的小振幅空间非均质项εF（U, Ux, x）驱动的显式、标量、双稳定Allen-Cahn方程的最简单设置。在第一部分中，我们分析了一般空间异质性F（U, Ux, x）的平稳一、二和n -锋模式的存在性和稳定性。然后，对于一般动态演化的n -锋模式，我们明确地确定了支配前沿位置演化的n阶ode系统。在第二部分中，我们将重点关注一类特殊的空间异质性，其中F(U,Ux,x)=H ' (x)Ux+H″(x)U，其中H要么是空间局部化的，要么是空间周期性的。对于局域异质性，我们确定了所有固定的N-锋模式，并表明这些模式对于N是不稳定的 >； 1。相反，我们发现由n个前沿组成的缓慢进化的“列车”共同向 ± ∞行进，要么速度缓慢降低，要么速度缓慢增加。对于空间周期性非均质性，我们表明，如果连续锋面之间的距离足够大，则多锋模式的锋面将被“钉住”，即多锋模式被附近稳定的静止多锋模式吸引。

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