Riemann–Hilbert Approach for the Hirota–Satsuma Coupled KdV System

ABSTRACT In this paper, the Riemann–Hilbert (RH) method is developed to solve the Hirota–Satsuma coupled KdV (HScKdV) system associated with matrix spectral problem. Because the spectral matrix is asymmetric and high order, it brings great difficulties to analysis and solution. First, a direct scattering problem is carried out, from which the initial data are mapped to the scattering data. On the basis of introducing the generalized cross product of vectors, the basic meromorphic matrix eigenfunctions of corresponding Lax pairs are expressed by the Jost solutions and adjoint Jost solutions. Then, the inverse scattering problem is characterized as the matrix RH problem, which gives the formula for constructing solutions of the HScKdV system. As an example, the RH problem corresponding to the reflectionless case is solved and the soliton solution of the HScKdV system is obtained.