Gerdjikov--Ivanov方程中非凸流体动力学下稀疏波与色散冲击波之间的相互作用

中文摘要 AI 辅助

本文通过在Gerdjikov--Ivanov（GI）方程中构造两步分段常数初始条件，系统研究了稀疏波（RW）与色散冲击波（DSW）之间相互作用的可容许构型。由于与Riemann不变量相关的双值映射，GI方程还支持接触色散冲击波（CDSWs），这些也被纳入相互作用分析中。利用Whitham调制理论，我们阐明了入射波的自相似描述，推导了边界匹配条件，并通过广义hodograph变换获得了相互作用前、中、后完整演化的解析表达式。结果表明，RW与DSW或CDSW的相互作用可以表现出两种不同的情形：一种情形是，在相互作用后，两个波交换谱参数并完全分离，以修正的相位独立传播；另一种情形是，波结构在相互作用时合并，随后演化为孤子列或小振幅谐波（对于经典DSW），而对于CDSW则演化为代数孤子列。理论结果与数值模拟吻合良好。这项工作可为非凸色散系统中的波相互作用现象提供理论支持。

英文摘要

This paper systematically investigates admissible configurations of the interaction between a rarefaction wave (RW) and a dispersive shock wave (DSW) by constructing a two-step piecewise constant initial condition in the Gerdjikov--Ivanov (GI) equation with non-convex hydrodynamics. Owing to the two-valued mapping associated with the Riemann invariants, the GI equation also supports contact DSWs (CDSWs), which are included in the interaction analysis. Using Whitham modulation theory, we elucidate the self-similar descriptions of the incoming waves, derive the boundary matching conditions, and obtain analytical expressions of the full evolution before, during, and after the interaction through generalized hodograph transformation. The results show that the interaction of a RW with either a DSW or a CDSW can exhibit two distinct scenarios: in one case, the two waves exchange spectral parameters after the interaction and fully separate, propagating independently with modified phases; in the other case, the wave structures merge upon interaction and subsequently evolve into either a soliton train or a small-amplitude harmonic wave for a classical DSW, and into an algebraic soliton train for a CDSW. The theoretical results are in good agreement with numerical simulations. This work can provide theoretical support for wave interaction phenomena in non-convex dispersive systems.

发表机构

• School of Mathematics, Taiyuan University（太原大学数学学院）
• Taiyuan 030024, China（中国太原）

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