Step-like initial value problem and Whitham modulation in fluid dynamics to a generalized derivative nonlinear Schrödinger equation

Abstract

In this paper, we study the step-like initial value problem for a generalized derivative nonlinear Schrödinger equation using the Whitham modulation theory. First, we utilize the finite-gap integration method to obtain the periodic solutions and the relevant Whitham equations for the 0-, 1-, and 2-genus cases used to characterize dispersive shock waves (DSWs). Second, we investigate four fundamental waves: two rarefaction waves (RWs) and two DSWs with step-like initial data. On this basis, we show the effect of varying certain parameters on the dynamics of the fluid model. We find that the boundary value, amplitude, and shape of the wave pattern in the fluid dynamics model will be significantly impacted by these parameters. Third, under two step-like initial data, waves are divided into six cases, which are actually combinations of DSWs and RWs. Finally, the dam break problem is explored to prove the effectiveness of the Whitham modulation theory in physical applications.