Analysis of wave propagation and conservation laws for a shallow water model with two velocities via Lie symmetry

IF 3.8 2区数学 Q1 MATHEMATICS, APPLIED Communications in Nonlinear Science and Numerical Simulation Pub Date : 2025-04-01 Epub Date: 2025-01-24 DOI:10.1016/j.cnsns.2025.108637

Aniruddha Kumar Sharma , Sumanta Shagolshem , Rajan Arora

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Abstract

This research investigates a one-dimensional system of quasi-linear hyperbolic partial differential equations, obtained by vertically averaging the Euler equations between artificial interfaces. This system represents a shallow water model with two velocities and is explored using Lie symmetry analysis to derive several closed-form solutions. Through symmetry analysis, a Lie group of transformations and its corresponding generators are identified via parameter analysis. From these, an optimal one-dimensional system of subalgebras is constructed and classified based on symmetry generators and invariant functions. The model is further simplified by reducing it to ordinary differential equations (ODEs) using similarity variables for each subalgebra, yielding invariant solutions. Additionally, various conservation laws are formulated utilizing the nonlinear self-adjointness property of the governing system. The study concludes by analyzing the behavior of characteristic shocks, C¹-waves, and their interactions, offering a detailed understanding of their dynamics.

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利用李氏对称分析两种速度浅水模型的波传播和守恒定律

本文研究了一类一维拟线性双曲型偏微分方程组，该方程组由人工界面间的欧拉方程垂直平均得到。该系统代表一个具有两种速度的浅水模型，并利用李氏对称分析得到了几个封闭形式的解。在对称性分析的基础上，通过参数分析确定了李群变换及其产生子。在此基础上，构造了一个最优的一维子代数系统，并基于对称生成和不变函数对其进行了分类。该模型通过对每个子代数使用相似变量将其简化为常微分方程（ode）进一步简化，从而得到不变解。此外，利用控制系统的非线性自伴随特性，建立了各种守恒律。该研究通过分析特征冲击、c1波及其相互作用的行为来总结，提供了对其动力学的详细理解。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.