The Well and Box Types Initial Value Problems of the defocusing mKdV Equation

In this paper, we study the time evolution problems of special initial values with two discontinuities for the defocusing mKdV equation. Based on the finite-gap integral method, Whitham modulation system of the defocusing mKdV equation is derived in the form of Riemann invariants, and full-time evolution results for six types of initial values with two discontinuities are obtained in detail. The regions where rarefaction waves interact with dispersive shock waves are thoroughly analyzed. Two novel phenomena, namely soliton trains and linear wave packets, are identified and their intrinsic mechanisms are systematically illustrated. All the results are verified by numerical simulations.