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

Abstract

AbstractThe 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.