Whitham modulation theory and exotic wave patterns of the good Jaulent–Miodek equation with step-like initial data

Abstract

The complete classification of solutions to the Riemann problem of the good Jaulent–Miodek equation is investigated by Whitham modulation theory. The one-phase and two-phase periodic wave solutions and the corresponding Whitham equations are derived based on the Lax pair of the good Jaulent–Miodek equation and the finite-gap integration method. In particular, the N-phase periodic wave solutions are proposed by algebro-geometric approach. Then the basic wave structures of rarefaction waves and DSWs are proposed analytically and graphically, which makes it possible to establish the classification of all the possible wave patterns evolving from initial discontinuities. The asymptotic results given by Whitham modulation theory are in excellent agreement with direct numerical simulations. Finally, a detailed description of the shallow-water dam break problem is demonstrated to find the possible physical significance of the wave patterns found in this work.