Inverse Scattering Transform for the Coupled Modified Complex Short Pulse Equation: Multiple Higher Order Poles Case

Abstract

We develop a Riemann–Hilbert (RH) approach to the inverse scattering transform for the coupled modified complex short pulse (cmcSP) equation when the reflection coefficient has multiple higher order poles. With the help of a generalized cross product defined in four-dimensional vector space, the discrete spectrum is specifically analyzed and the $4\times 4$ RH problem with generalized residue conditions at the $N$ pairs of higher order poles is constructed. In the reflectionless case, the RH problem can be reduced to a linear algebraic system. Consequently, the general formulas for the corresponding multiple higher order pole solutions of the cmcSP equation are obtained. The dynamical behavior of several pole-type solutions is exhibited, including one double-pole solutions (smooth solitons, cuspons, breathers), and several collisions between one double-pole solution and one simple pole solution.