Riemann-Hilbert method and multi-soliton solutions of an extended modified Korteweg-de Vries equation with N distinct arbitrary-order poles

Abstract

The Riemann-Hilbert (RH) method is developed to study the extended modified Korteweg-de Vries (emKdV) equation with zero boundary conditions. The analytical and asymptotic properties of Jost functions are obtained by the direct scattering analysis to establish a suitable RH problem. We consider the singular RH problem of scattering data with N distinct poles. By using the generalized residue condition to solve the RH problem, we construct the exact solution of emKdV equation under the condition of no reflection. In addition, four kinds of special poles and their corresponding soliton solutions are discussed in detail, including one second-order pole, one third-order pole, three first-order poles and two second-order poles.