Solving the modified Camassa–Holm equation via the inverse scattering transform

With the aid of the reciprocal transformation and the associated equation, we study the inverse scattering transform with a matrix Riemann–Hilbert problem for the modified Camassa–Holm (mCH) equation with nonzero boundary conditions (NZBC) at infinity. In terms of a suitable uniformization variable, the direct and inverse scattering problems are presented for the associated modified Camassa–Holm (amCH) equation. By means of the reciprocal transformation and the reconstruction formula for the potential of the amCH equation, we present the $N$-soliton solution for the mCH equation with NZBC. As applications, various solutions including both bright and dark types, smooth soliton solutions, singular soliton solutions, and multi-valued singular soliton solutions of the mCH equation and their interactions are exhibited.