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 : 2026-01-28 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

Center manifold theorem of fractional differential equations and machine learning under weak data 分数阶微分方程中心流形定理与弱数据下的机器学习

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-01-24 DOI: 10.1016/j.physd.2026.135122

Han-Lin Liao, Guo-Cheng Wu, Dong Li

Fractional differential equations frequently arise in long-range interaction processes. The center manifold theorem is an essential tool in reduction of dynamical systems. First, this paper provides existence conditions for center manifolds by constructing function spaces and fixed-point mappings. Then, determining the center manifolds becomes a parameter estimation problem. Because the chain rule for fractional derivatives cannot be applied, a neural network method is developed to find approximate center manifolds near the zero equilibrium. The automatic model selection is employed to search for a neural network architecture. Two examples are presented to demonstrate the efficiency of reducing high-dimensional fractional order systems under weak data.

在远距离相互作用过程中经常出现分数阶微分方程。中心流形定理是研究动力系统约简的重要工具。首先，通过构造函数空间和不动点映射，给出了中心流形的存在条件。然后，中心流形的确定就变成了一个参数估计问题。由于分数阶导数的链式法则不能应用，提出了一种神经网络方法来求零平衡点附近的近似中心流形。采用自动模型选择方法搜索神经网络结构。给出了两个例子来证明在弱数据下高维分数阶系统的约简效率。

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 : 2026-01-24 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

Recent progress on coarse graining simulations 粗粒化模拟研究进展

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-01-22 DOI: 10.1016/j.physd.2026.135116

Fernando F․ Grinstein, Vincent P․ Chiravalle, Robert K. Greene

We focus on coarse graining simulations based on the primary conservation equations, effectively codesigned physics and algorithms, and low-Mach-number corrected (LMC) hydrodynamics. Simulation methods involve LANL’s x-Radiation-Adaptive-Grid-Eulerian Large-Eddy Simulation, Besnard-Harlow-Rauenzahn (BHR) Reynolds-Averaged Navier-Stokes (RANS) approach, and Dynamic BHR – a paradigm bridging RANS and LES.

A relevant question addressed relates to whether 3D RANS and RANS/LES hybrids – the industry standards for aerospace and automotive research, are presently relevant for practical variable-density applications involving shocked and accelerated interface instabilities. Recent simulations of the GaTECH inclined mixing-layer shock-tube and NIF ICF-capsule experiments are used to demonstrate issues, challenges, and potential for 3D coarse grained LMC simulation strategies for robustly simulating complex transitional and coupled hydrodynamics-multiphysics with coarser resolution. Present LES readiness to provide accurate predictions at scale is demonstrated – whereas 3D RANS and RANS/LES bridging do not appear impactful in this context.

我们的重点是基于初级守恒方程的粗粒模拟，有效的协同设计物理和算法，以及低马赫数校正（LMC）流体动力学。模拟方法包括LANL的x-辐射自适应网格-欧拉大涡模拟，Besnard-Harlow-Rauenzahn (BHR) reynolds - average Navier-Stokes （RANS）方法，以及Dynamic BHR -一种连接RANS和LES的范式。一个相关的问题是，3D RANS和RANS/LES混合动力系统——航空航天和汽车研究的行业标准，目前是否适用于涉及冲击和加速界面不稳定性的实际变密度应用。最近对GaTECH倾斜混合层激波管和NIF icf胶囊实验的模拟显示了3D粗粒度LMC模拟策略的问题、挑战和潜力，这些策略可以以更粗的分辨率鲁棒地模拟复杂的过渡和耦合流体动力学-多物理场。目前的LES已经准备好提供大规模的准确预测，而3D RANS和RANS/LES桥接在这种情况下似乎没有影响。

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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 : 2026-01-21 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

Slip topology of three-dimensional homogeneous quadratic velocity fields 三维齐次二次速度场的滑移拓扑

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-01-19 DOI: 10.1016/j.physd.2026.135121

Wennan Zou, Jian He

The contact structures of fluid are described by the streamline pattern in steady flows, where the key to determine the slip topology the streamline pattern around the isotropic point, called the local streamline pattern (LSP). In this paper, taking homogeneous quadratic velocity fields (HQVFs) as the research object and utilizing the swirl field, which is an axis-vector-valued differential 1-form determined by the velocity direction, to define the topological degree, we establish an analytical framework for three-dimensional nonlinear velocity fields. After obtaining the trivial result of the topological degree of three-dimensional HQVFs, we make use of the characteristic problems of high order tensor to work out all radial streamlines entering/exiting an isotropic point, and adopt the pair number of radial streamlines as the key criterion to classify the LSPs. Some typical HQVFs are illustrated for discussion, and the investigation on linear velocity fields shows their particularity. As a preliminary exploration of the streamline pattern of three-dimensional nonlinear velocity fields, this work demonstrates how difficult it is to generalize the research results of two-dimensional velocity fields and three-dimensional linear velocity fields.

流体的接触结构用定常流动中的流线模式来描述，其中确定滑移拓扑的关键是各向同性点周围的流线模式，称为局部流线模式（LSP）。本文以齐次二次速度场（HQVFs）为研究对象，利用由速度方向决定的轴向值微分1型旋流场来定义拓扑度，建立了三维非线性速度场的解析框架。在得到三维HQVFs拓扑度的平凡结果后，利用高阶张量的特征问题求出了进出各向同性点的所有径向流线，并以径向流线的对数作为对lsp进行分类的关键准则。举例说明了一些典型的HQVFs，对线速度场的研究显示了它们的特殊性。作为对三维非线性速度场流线模式的初步探索，本工作证明了将二维速度场和三维线速度场的研究成果推广是多么困难。

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引用次数: 0

Darboux transformations and related non-Abelian integrable differential-difference systems of the derivative nonlinear Schrödinger type 导数非线性Schrödinger型的达布变换及相关非阿贝尔可积微分-差分系统

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-01-18 DOI: 10.1016/j.physd.2026.135119

Edoardo Peroni , Jing Ping Wang

We construct linear and quadratic Darboux matrices compatible with the reduction group of the Lax operator for each of the seven known non-Abelian derivative nonlinear Schrödinger equations that admit Lax representations. The differential-difference systems derived from these Darboux transformations generalise established non-Abelian integrable models by incorporating non-commutative constants. Specifically, we demonstrate that linear Darboux transformations generate non-Abelian Volterra-type equations, while quadratic transformations yield two-component systems, including non-Abelian versions of the Ablowitz-Ladik, Merola-Ragnisco-Tu, and relativistic Toda equations. Using quasideterminants, we establish necessary conditions for factorising a higher-degree polynomial Darboux matrix with a specific linear Darboux matrix as a factor. This result enables the factorisation of quadratic Darboux matrices into pairs of linear Darboux matrices.

对于七个已知的允许Lax表示的非阿贝尔导数非线性Schrödinger方程，我们构造了与Lax算子约化群相容的线性和二次Darboux矩阵。由这些达布变换导出的微分-差分系统通过引入非交换常数来推广已建立的非阿贝尔可积模型。具体来说，我们证明了线性Darboux变换产生非abelian volterra型方程，而二次变换产生双组分系统，包括Ablowitz-Ladik， Merola-Ragnisco-Tu和相对论Toda方程的非abelian版本。利用拟行列式，建立了以特定线性达布矩阵为因式分解高次多项式达布矩阵的必要条件。这个结果使二次达布矩阵分解成线性达布矩阵对成为可能。

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 : 2026-01-17 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

Bose-Einstein condensates with density-dependent gauge potential and two PT-symmetric potentials: Solitons, rogue waves and nonlinear dynamics 具有密度依赖规范势和两个pt对称势的玻色-爱因斯坦凝聚体：孤子、异常波和非线性动力学

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-01-16 DOI: 10.1016/j.physd.2026.135117

Yufeng Liu , Jin Song , Zhenya Yan

In this paper, we investigate the dynamical behaviors of soliton solutions and rogue waves in Bose-Einstein condensates, modelled by the focusing and defocusing Gross-Pitaevskii equations. Our work uniquely integrates the effects of a density-dependent gauge potential with two novel types of parity-time (

PT

) symmetric potentials: the generalized

PT

-symmetric harmonic-Gaussian potential and the generalized

PT

-symmetric Scarf-II potential. This specific combination provides a rich platform for exploring the phenomenon of matter waves. Firstly, we analyze the linear spectral problem to determine the entirely real spectral region of the non-Hermitian Hamiltonian and examine the occurrence of the phase-breaking phenomena. Then, we discuss the exact solutions for both types of

PT

-symmetric external potentials, as well as the numerical solutions for the ground state and dipole modes, while analyzing their stability using the corresponding Bogoliubov-de Gennes equations. Moreover, we investigate the effect of the current nonlinearity on the stability of exact solitons. In particular, the interactions between two solitons are studied, exhibiting nearly elastic and inelastic interactions. Furthermore, the stable adiabatic excitations of solitons are investigated. Finally, due to the influence of current nonlinearity on its structure, the high-order rogue wave generated by several Gaussian perturbations on the continuous wave undergoes a transformation into chiral solitons with lower amplitude. Our research provides in-depth insights into the dynamic behavior of solitons and rogue waves in novel systems with density-dependent gauge potential and

PT

-symmetric external potential, offering important guidance for future theoretical research and experimental exploration of complex matter wave phenomena.

在本文中，我们研究了玻色-爱因斯坦凝聚体中孤子解和异常波的动力学行为，用聚焦和散焦Gross-Pitaevskii方程来模拟。我们的工作独特地整合了密度相关规范势与两种新型的奇偶时间对称势的影响：广义PT对称谐波高斯势和广义PT对称Scarf-II势。这种特殊的组合为探索物质波现象提供了丰富的平台。首先，我们分析了线性谱问题，确定了非厄米哈密顿量的全实数谱区，并检验了破相现象的发生。然后，我们讨论了两种pt对称外势的精确解，以及基态和偶极子模式的数值解，同时使用相应的Bogoliubov-de Gennes方程分析了它们的稳定性。此外，我们还研究了电流非线性对精确孤子稳定性的影响。特别地，研究了两个孤子之间的相互作用，表现出近弹性和非弹性相互作用。进一步研究了孤子的稳定绝热激发。最后，由于电流非线性对其结构的影响，多次高斯扰动对连续波产生的高阶异常波转变为振幅较低的手性孤子。我们的研究深入了解了具有密度依赖规范势和pt对称外势的新型系统中孤子和异常波的动力学行为，为未来复杂物质波现象的理论研究和实验探索提供了重要指导。

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引用次数: 0

Analysis of traveling wave solutions of a qualitative model of combustion waves 燃烧波定性模型的行波解分析

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena

Pub Date : 2026-01-16 DOI: 10.1016/j.physd.2026.135114

Shamil M. Magomedov, Aslan R. Kasimov

We investigate the structure of steady-state traveling-wave solutions of a model system of partial differential equations that describes combustion waves of both subsonic and supersonic type. The solutions are found both theoretically by employing matched asymptotic expansions in the small-dissipation limit and numerically. The existence of various types of traveling waves is shown, which include detonation-like as well as fast and slow deflagration-like solutions. The model was originally introduced to mimic properties of Navier–Stokes equations for reactive compressible flows, and here we demonstrate how closely these systems are related.

我们研究了一个描述亚音速和超音速燃烧波的偏微分方程模型系统的稳态行波解的结构。利用小耗散极限下的匹配渐近展开式在理论上和数值上得到了这些问题的解。证明了各种类型行波的存在，包括类爆轰解以及快速和慢速类爆燃解。该模型最初是为了模拟反应性可压缩流的Navier-Stokes方程的性质而引入的，在这里我们展示了这些系统是如何紧密相关的。

引用次数: 0

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