阶跃初值条件解的完全分类: 流体力学中广义 Gardner 方程的分析与数值验证

Abstract

Abstract:

In this paper, we investigate the evolution of the initial discontinuity for the generalized Gardner equation through the Whitham modulation theory, which the generalized Gardner equation can describe the transcritical flow of stratified fluids over topography. Firstly, we derive the linear harmonic wave, soliton and nonlinear trigonometric wave in different limiting cases via the periodic waves represented by the Jacobi elliptic functions. Then we obtain the Whitham characteristic velocities and modulation system based on the Riemann invariants by the finite-gap integration method. Since the modulation system of the generalized Gardner equation is neither strictly elliptic nor hyperbolic type, which makes the dynamical evolution behavior more varied in different regions compared to the KdV equation. Furthermore, we perform a complete classification for all wave structures in the cases of positive and negative cubic nonlinear terms, including the dispersive shock wave, rarefaction wave, trigonometric dispersive shock wave, solibore and their combinations. In addition, the correctness of the results is verified by numerical simulations, and the numerical solutions are in good agreement with the analytical solutions. Finally, the influences of the coefficients of the linear and nonlinear terms on the step initial value problem under certain conditions are analyzed.

Key words: Generalized Gardner equation, Finite-gap integration method, Riemann invariant, Whitham modulation theory, Initial discontinuity, Dispersive shock wave

CLC Number:

O322

Zhang Yan, Hao Huiqin, Guo Rui. The Complete Classification of Solutions to the Step Initial Condition: Analysis and Numerical Verification for the Generalized Gardner Equation in Fluid Mechanics[J].Acta mathematica scientia,Series A, 2024, 44(5): 1242-1282.

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Figures/Tables 47

References 55

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The Complete Classification of Solutions to the Step Initial Condition: Analysis and Numerical Verification for the Generalized Gardner Equation in Fluid Mechanics

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