Riemann-Hilbert problem and multiple poles solution for an extended modified Korteweg-de Vries equation with zero/nonzero boundary conditions

The aim is to present the inverse scattering transform (IST) with Riemann-Hilbert problem (RHP) for a higher-order extended modified Korteweg-de Vries (emKdV) equation with zero/nonzero boundary conditions (Z/NZBC) at infinity, where the emKdV equation contains third- and fifth-order dispersion coefficients matching with relevant nonlinearity terms, respectively. Under reflectionless condition, the RHP of emKdV equation is firstly solved when scattering data have two cases: multiple simple poles and higher-order poles, and corresponding formulae of multiple simple poles soliton and higher-order poles solution are shown in terms of determinants, respectively. One simple soliton and two simple soliton solutions are obtained in detail according to variable values of third- and fifth-order dispersion coefficients, the effect power of which are specially displayed in dynamic structures. In addition, as a very important application for solving higher-order solutions by RHP, we extend Laurent’s series of residue conditions of idea to this higher-order emKdV equation.