Discontinuous initial value and Whitham modulation for the generalized Gerdjikov-Ivanov equation

Abstract

This paper mainly focuses on the possible wave patterns for the discontinuous initial value problem of the generalized Gerdjikov-Ivanov equation utilizing the Whitham modulation theory. The zero-phase, one-phase and two-phase solutions of the generalized Gerdjikov-Ivanov equation along with the corresponding Whitham equations are constructed by the approach of finite-gap integration. Considering the self-similar solution of the Whitham equations, the elementary wave patterns of rarefaction wave, cnoidal dispersive shock wave, contact dispersive shock wave and combined shocks are found. For the general step-like condition of initial discontinuity, the evolution wave patterns consist of plateau, rarefaction wave and cnoidal dispersive shock wave. The classification of all possible wave patterns in the monotonic region is presented graphically. Furthermore, the results of the Whitham modulation theory are verified by full numerical simulations.