Evolution of dispersive shock waves to the complex modified Korteweg–de Vries equation with higher-order effects

Abstract

In this paper, new dispersive shock waves (DSWs) in step-like initial value problems to the complex modified Korteweg–de Vries (cmKdV) equation with higher-order effects are found via Whitham modulation theory. For the aforementioned equation, the 1-genus and 2-genus periodic solutions and the associated Whitham equations which are used to describe DSWs are firstly given by the finite-gap integration method, and we also analyze nine types of rarefaction waves appearing before DSWs under the 0-genus Whitham equations. Subsequently, the DSW solutions with step-like initial data are discussed, where we acquire some DSW structures that have not been previously proposed. These notable new results include 1-genus DSW satisfying that one Riemann invariant is constant and the other three are variables and 2-genus DSW in the DSW solutions with one step-like initial data, as well as 3-genus DSW resulting from the collision to 1-genus and 2-genus or two 2-genus DSWs propagating toward each other in the possible DSW solutions with two step-like initial data. Ultimately, the dam break problem is explored to demonstrate the significant physical application of the theoretical findings.