On transverse spectral instabilities to the (2+1)-dimensional Boussinesq equation

IF 2.9 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena Pub Date : 2025-08-25 DOI:10.1016/j.physd.2025.134891

Wen-Wu Zhou, Shou-Fu Tian

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Abstract

The primary objective of this study is to explore the spectral stability of one-dimensional small-amplitude periodic traveling wave solutions for the two-dimensional Boussinesq equation. This investigation offers a framework for comprehending intricate wave interactions across a diverse range of fluid systems and underscores the interaction between nonlinearity and dispersion during wave propagation. Through the analysis of the associated spectral problem, we discover that these periodic traveling waves are unstable under long-wavelength perturbations in both transverse directions. This finding implies that small disturbances can induce substantial alterations in wave propagation. Moreover, we demonstrate that perturbations that are periodic or square-integrable with zero mean in wave propagation, along with finite or short-wavelength periodic perturbations in the transverse direction, display stability. Our results establish the specific conditions under which transverse stability is ensured, thereby highlighting the significance of perturbation characteristics in determining the stability of wave solutions within the context of shallow water wave theory.

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（2+1）维Boussinesq方程的横向谱不稳定性

本研究的主要目的是探讨二维Boussinesq方程的一维小振幅周期行波解的谱稳定性。这项研究为理解不同流体系统中复杂的波相互作用提供了一个框架，并强调了波传播过程中非线性和色散之间的相互作用。通过对相关光谱问题的分析，我们发现这些周期行波在两个横向的长波摄动下都是不稳定的。这一发现意味着微小的扰动可以引起波传播的重大变化。此外，我们证明了在波传播中周期或平方可积与零平均值的扰动，以及在横向上有限波长或短波长的周期性扰动，显示出稳定性。我们的结果建立了确保横向稳定性的具体条件，从而突出了在浅水波浪理论背景下确定波浪解稳定性的摄动特征的重要性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Physica D: Nonlinear Phenomena 物理-物理：数学物理

CiteScore

7.30

自引率

7.50%

发文量

213

审稿时长

65 days

期刊介绍： Physica D (Nonlinear Phenomena) publishes research and review articles reporting on experimental and theoretical works, techniques and ideas that advance the understanding of nonlinear phenomena. Topics encompass wave motion in physical, chemical and biological systems; physical or biological phenomena governed by nonlinear field equations, including hydrodynamics and turbulence; pattern formation and cooperative phenomena; instability, bifurcations, chaos, and space-time disorder; integrable/Hamiltonian systems; asymptotic analysis and, more generally, mathematical methods for nonlinear systems.