Inverse scattering problem for the matrix modified Korteweg–de Vries equation with finite density type initial data

Abstract

The theory of inverse scattering is developed to study the initial-value problem for the matrix modified Korteweg–de Vries (mKdV) equation with the 2m×2m(m≥1) Lax pairs and finite density type initial data. In direct scattering problem, by introducing a suitable uniform transformation we establish the proper complex z-plane in order to discuss the Jost eigenfunctions, scattering matrix and their analyticity and symmetry. Moreover, the asymptotic behavior of the Jost eigenfunctions and scattering matrix are analyzed needed in inverse problem. Then, the generalized Riemann–Hilbert problem of the matrix mKdV equation is established by using the analyticity of the modified eigenfunctions and scattering coefficients, from which the reconstruction formula of potential function with reflection-less case is derived successfully. Finally, some interesting phenomena in scalar and matrix cases are presented, as well as theoretical analysis of asymptotic behavior of solutions in a appendix.