LAUSR

Abstract In this paper, we study the inverse scattering transform for the coupled modified Korteweg-de Vries (cmKdV) equation with nonzero boundary conditions (NZBCs) at infinity. In order to deal with the direct and inverse scattering problems, a suitable uniformization variable is given on a complex plane by transforming a two-sheeted Riemann surface. In the direct scattering problem, we discuss the analyticity, three symmetries and asymptotic behaviors of the Jost functions, scattering matrices and the distribution of discrete spectral. In the inverse scattering problem, the Riemann-Hilbert (RH) problem of cmKdV equation is first found by using the analyticity of the modified eigenfunctions and scattering coefficients. The reconstruction formulae solution with simple poles, trace formulae and asymptotic phase difference are obtained. Finally, we analyze soliton solutions for eight distinct wave structures of reflectionless potential in the RH problem, include one-kink, dark, bright solitons, novel periodic solutions and two-soliton solutions.