Exotic wave patterns in Riemann problem of the high-order Jaulent-Miodek equation: Whitham modulation theory

The Riemann problem of the high-order Jaulent-Miodek (JM) equation with initial data of step discontinuity is explored by Whitham modulation theory, which is a modified version of the well-known finite-gap integration method. Based on the reparameterization of the solution with the use of algebraic resolvent of the polynomial defining the solution, the periodic wave solutions of the high-order JM equation are described by the elliptic function along with the Whitham modulation equations. Complete classification of possible wave structures of the high-order JM equation is given for all possible jump conditions at the discontinuity initial value. The analytic results proposed in this work are confirmed by direct numerical simulations.