广义kap - boussinesq方程不连续初值问题的Whitham调制理论

IF 2.7 3区数学 Q1 MATHEMATICS, APPLIED

Physica D: Nonlinear Phenomena Pub Date : 2025-02-14 DOI:10.1016/j.physd.2025.134573

Ruizhi Gong, Deng-Shan Wang

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引用次数: 0

摘要

利用Whitham调制理论研究了能模拟浅水波浪运动现象的广义kap - boussinesq （KB）方程不连续初值问题解的完全分类。根据色散关系，广义KB方程分别包括广义good-KB方程和广义bad-KB方程。首先，利用Flaschka-Forest-McLaughlin方法给出了与广义bad-KB方程相关的周期波解和相应的Whitham方程。其次，通过分析零属Whitham方程和一属Whitham方程，描述了基本的稀疏波结构和色散激波结构。然后给出广义bad-KB方程的Riemann问题解的完全分类，并对18种不同的情况进行了分类，其中包括5种临界情况。详细地证明了黎曼不变量的分布和各分量的自相似态的演化。结果表明，精确的孤子解与调制色散激波的孤子边缘符合得很好。此外，还观察到，在每种情况下，相位肖像与调制解的行为建立了一致的关系。最后，对于广义的good-KB方程，探讨了一类新的具有常周期波边界的不连续初值问题，并得到了一些新的具有三角激波的调制解。指出在广义bad-KB方程中不存在这种三角激波，因为周期波的小振幅极限不是三角函数而是常数。研究结果揭示了浅水中奇异的破波现象，为研究非线性色散方程的不连续初值问题提供了一种可行的方法。

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Whitham modulation theory for the discontinuous initial-value problem of the generalized Kaup–Boussinesq equation

The Whitham modulation theory is developed to investigate the complete classification of solutions to discontinuous initial-value problem of the generalized Kaup–Boussinesq (KB) equation, which can model phenomenon of wave motion in shallow water. According to the dispersion relation, the generalized KB equation includes the generalized good-KB equation and generalized bad-KB equation, respectively. Firstly, the periodic wave solutions and the corresponding Whitham equations associated with the generalized bad-KB equation are given by Flaschka–Forest–McLaughlin approach. Secondly, the basic rarefaction wave structure and dispersive shock wave structure are described by analyzing the zero-genus and one-genus Whitham equations. Then the complete classification of solutions to Riemann problem of the generalized bad-KB equation is provided, and eighteen different cases are classified, including five critical cases. The distributions of Riemann invariants and the evolutions of self-similar states for each component are demonstrated in detail. It is shown that the exact soliton solution is in good agreement with the soliton edge of the modulated dispersive shock wave. Moreover, it is observed that the phase portraits in each case establish a consistent relationship with the behavior of the modulated solutions. Finally, for the generalized good-KB equation, a new type of discontinuous initial-value problem with constant-periodic wave boundaries is explored, and some novel modulated solutions with trigonometric shock waves are found. It is remarked that such trigonometric shock waves are absent in the generalized bad-KB equation because the small amplitude limits of the periodic waves are not trigonometric functions but constants. The results in this work reveal exotic wave-breaking phenomena in shallow water and provide a feasible way to investigate the discontinuous initial-value problem of nonlinear dispersive equations.

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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.