Wave motions in discontinuous initial-value problem of the inviscid shallow water wave Jaulent-Miodek model

The wave motions of the initial discontinuities in the inviscid shallow water wave model governed by the bad Jaulent-Miodek system are explored by the Whitham modulation theory and dynamical analysis. The N-phase algebro-geometric solutions along with the related genus-N Whitham equations are derived by algebro-geometric method and Flaschka-Forest-McLaughlin approach, respectively. The complete classification of solutions to the bad Jaulent-Miodek system is carried out by solving the Whitham equations both theoretically and numerically. Some exotic phenomena of nonlinear wave motions such as rarefaction waves and dispersive shock waves are found in the inviscid shallow water. Particularly, the nonlinear wave motions including the two-phase algebro-geometric solutions are found in the inviscid shallow water wave. It is shown that the results from Whitham modulation theory agree very well with the numerical calculations and dynamical analysis.