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

Yuan Xiang; Rui Guo

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arXiv · 2608.03139

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

Yuan Xiang·Rui Guo

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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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Yuan Xiang, Rui Guo. 2026-08-04. The co-directional overtaking collision problem between a dispersive shock wave and a rarefaction wave for the Hirota equation. https://arxiv.org/abs/2608.03139

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