Evolution of initial discontinuities in the Riemann problem for the Jaulent–Miodek equation with positive dispersion

Abstract

On the basis of Whitham modulation theory, we consider the complete classification of solutions of the initial discontinuities for the Jaulent–Miodek (JM) equation with positive dispersion. The Whitham equations of the JM system are obtained with the finite-gap integration theory, in which dispersive shock waves (DSW) can be described with one-phase modulated traveling wave solution. Since the JM system with positive dispersion is modulationally unstable, the corresponding behaviors of Riemann problem are more complicated than the stable JM system. All the possible wave configurations evolving from initial discontinuities are combinations of the plateau, vacuum, rarefaction wave, DSW, as well as overlap of rarefaction wave and DSW. The results of JM-Whitham system are applied to describe the wave structure in physical model cases, the dam breaking problem and piston problem.