Huang Ya-Hui, Guo Rui
School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China.
Chaos. 2024 Oct 1;34(10). doi: 10.1063/5.0231741.
We study the problem of wave breaking for a simple wave propagating to a quiescent medium in the framework of the defocusing complex modified KdV (cmKdV) equation. It is assumed that a cubic root singularity is formed at the wave-breaking point. The dispersive regularization of wave breaking leads to the generation of a dispersive shock wave (DSW). We describe the DSW as a modulated periodic wave in the framework of the Gurevich-Pitaevskii approach based on the Whitham modulation theory. The generalized hodograph method is used to solve the Whitham equations, and the boundaries of the DSW are found. Most importantly, we determine the correct phase shift for the DSW from the generalized phase relationships and the modified Gurevich-Pitaevskii matching conditions, so that a complete description of the DSW is obtained rather than just its envelope. All of our analytical predictions agree well with the numerical simulations.
我们在散焦复修正KdV（cmKdV）方程的框架下，研究简单波向静态介质传播时的波破裂问题。假设在波破裂点形成了一个立方根奇点。波破裂的色散正则化导致了色散激波（DSW）的产生。我们基于惠特姆调制理论，在古列维奇 - 皮塔耶夫斯基方法的框架内，将DSW描述为调制周期波。使用广义速端曲线法求解惠特姆方程，并确定了DSW的边界。最重要的是，我们从广义相位关系和修正的古列维奇 - 皮塔耶夫斯基匹配条件中确定了DSW的正确相移，从而获得了对DSW的完整描述，而不仅仅是其包络。我们所有的解析预测与数值模拟结果都非常吻合。
