AI 中文总结
本文在Gardner方程框架下，分析凸与非凸情形下稀疏波与弥散激波的相互作用，求解Gardner-Whitham方程，明确不同α取值下的相互作用构型及演化结果。
AI 中文摘要
本文在Gardner方程框架内，重点分析了跨越凸性与非凸性情形的稀疏波与弥散激波之间相互作用的解析描述，尤其关注非凸性的影响。对于凸结构，当t趋于无穷时，内部振荡退化为小振幅谐波或调制孤子序列，伴随保留的稀疏波部分，或从中产生新的弥散激波。考虑非凸性，当α>0时，我们发现扭结要么完全不参与相互作用，要么仅用于切换凸结构的极性；当α<0时，我们求解具有三个变化黎曼不变量的Gardner-Whitham方程，分析稀疏-接触弥散激波的相互作用，其中唯一可能的构型是稀疏波位于左侧。结果表明，稀疏波将被完全吸入相互作用区域，改变后的接触弥散激波从左侧逸出，且当t趋于无穷时，内部振荡最终退化为渐近代数孤子序列。此外，我们在两种不同情形下研究了稀疏波与由接触和经典弥散激波部分组成的复合结构之间的相互作用：（i）对于复合结构-稀疏波相互作用，接触部分在相互作用中保持不活跃，内部振荡最终退化为接触弥散激波；（ii）对于稀疏波-复合结构情形，整个复合结构参与相互作用，在此过程中
英文摘要
In this paper, we focus on the analytical description of the interaction between a rarefaction and dispersive shock wave across both convex and non-convex cases, with particular attention to the effect of the non-convexity, within the framework of the Gardner equation. For convex structures, internal oscillations degenerate into small amplitude harmonic waves or a modulated soliton train as t tends to infinity, accompanied by either a retained rarefaction part, or a new dispersive shock wave emanating from it. Taking into account the non-convexity, when alpha>0, we find that kinks either remain non-participating in the interaction at all, or only act to switch polarities of convex structures. As for alpha<0, we solve the Gardner-Whitham equations with three varying Riemann invariants, to analyze the rarefaction-contact dispersive shock interaction where the only possible configuration is that the rarefaction wave is on the left. It is demonstrated that the rarefaction wave will be completely drawn into the interaction region, with a changed contact dispersive shock wave escaping from the left. And internal oscillations eventually degenerate into an asymptotic algebraic soliton train as t tends to infinity. In addition, we study the interaction between a rarefaction wave and composite structure consisting of the contact and classical dispersive shock parts under two distinct situations: (i) For the composite structure-rarefaction interaction, the contact part remains inactive in the interaction, and internal oscillations ultimately degenerate into a contact dispersive shock wave. (ii) For the rarefaction-composite structure case, the entire composite structure participates in the interaction, during
