Complete classification of solutions to the Riemann initial value problem for the Hirota equation with weak dispersion term

Jing Chen, Erbo Li, Yushan Xue

Abstract: In this paper, the Riemann problem for the defocusing Hirota equation with weak dispersion is investigated with Whitham modulation theory. The Hirota equation can effectively describe the realistic wave motion in a dispersive medium. Via the averaging Lagrangian method, the Whitham modulation equations in slow modulation form are obtained, which are characterized by wave parameters and reflect the dispersion relation in the original system. Besides, the modulation equations inRiemann invariant form are derived via finite-gap integration theory. UtilizingWhitham modulation equations parameterized by Riemann invariants, the basic structures of solutions for the Riemann problem for the original system are acquired. According to the basic structure of the solutions, a complete solution classification corresponding to the initial data is given, including 121 categories. The results are verified by direct numerical simulation.