The co-directional overtaking collision problem between a dispersive shock wave and a rarefaction wave for the Hirota equation

2026/08/04 by Yuan Xiang, Rui Guo
Physics and Astronomy · Mathematics · #math-ph #math.MP

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arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

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.

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