Riemann–Hilbert problem for the generalized modified Korteweg–de Vries equation with N distinct arbitrary-order poles

Abstract

In this work, the generalized modified Korteweg–de Vries (gmKdV) equation is constructed by the first time and is solved by the Riemann–Hilbert method with the zero boundary condition. In the direct scattering transform, the analytical and asymptotic properties related to the Jost functions and the scattering matrix are given. On the basis of the above results, the appropriate Riemann–Hilbert problem (RHP) is constructed. By solving the RHP, we obtain the exact solution of the gmKdV equation in the case of no reflection potential when the scattering data [Formula: see text] has simple poles and higher-order poles. Furthermore, the three special solutions under different zero points are given and the phenomenon of their spread is described, respectively.