Hirota方程中色散激波与稀疏波的同向超车碰撞问题
The co-directional overtaking collision problem between a dispersive shock wave and a rarefaction wave for the Hirota equation
AI总结：
本文研究散焦Hirota方程框架下同向传播的色散激波与稀疏波的超车碰撞问题，基于Whitham理论构造初始构型，通过广义hodograph变换等得到碰撞域调制解，解析描述碰撞动力学并经数值模拟验证，为该类碰撞问题提供了详细的理论分析。
AI中文摘要：
本文主要研究在散焦Hirota方程框架下，同向传播的色散激波（DSW）与稀疏波（RW）之间的超车碰撞问题。基于Whitham理论，给出了对应于散焦Hirota方程的零相位和单相位调制系统，以及黎曼不变量的显式表达式。对于左侧为DSW、右侧为RW且传播方向相同的阶跃型初始条件，构造了全局初始构型。通过广义 hodograph变换和欧拉-泊松-达布方程的解，得到了非线性碰撞域内的调制解。因此，通过匹配DSW和RW边界处的黎曼不变量，可对碰撞在所有演化阶段的动力学行为给出详细的解析描述。通过与直接数值模拟结果对比，验证了理论预测的准确性。
英文摘要：
In this paper, we mainly investigate the overtaking collision problem between a dispersive shock wave (DSW) and a rarefaction wave (RW) propagating in the same direction in the defocusing Hirota equation framework. Based on the Whitham theory, the zero-phase and single-phase modulation systems corresponding to the defocusing Hirota equation are given, along with explicit expressions of Riemann invariants. For step-type initial conditions with left-side DSW and right-side RW propagating identically, the global initial configuration is constructed.~The modulation solution within the nonlinear collision domain is obtained via the generalized hodograph transformation and solutions of the Euler-Poisson-Darboux equation. Thus, by matching the Riemann invariants at the boundaries of DSW and RW, we can provide a detailed analytical description for the dynamical behaviors of the collision across all evolutionary stages. The accuracy of theoretical predictions is verified by comparison with direct numerical simulations.
