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

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.