Exact multi-soliton solutions of the KdV equation with a source: Riemann-Hilbert formulation

The soliton solutions with non-zero background of KdV equation with a source are constructed by Riemann-Hilbert approach. The irregular Riemann-Hilbert problem is constructed by direct and inverse scattering transform firstly, and then it can be regularized by introducing a novel transformation. The Residue theorem is utilized to derive the multi-soliton solutions at the simple poles of the Riemann-Hilbert problem. In particular, the interaction dynamics of the two-soliton solution are illustrated by considering their evolutions at different time.