Riemann–Hilbert approach for a (2+1) dimensional Kundu–Mukherjee–Naskar equation

This paper concentrates on a (2+1) dimensional Kundu–Mukherjee–Naskar (KMN) equation. The (2+1) dimensional KMN equation was decomposed into two (1+1)-dimensional nonlinear evolution equations, resulting in the emergence of three spectral matrices. From these three spectral matrices, the Riemann–Hilbert problem of the (2+1) dimensional KMN equation is constructed. By solving the Riemann–Hilbert problem in the case that the jump matrix is identity matrix, that is, when the scattering data \(s_{12}(\lambda )\) $s_{12}(\lambda )$ and \(s_{21}(\lambda )\) $s_{21}(\lambda )$ are 0, the multi-soliton solutions for the (2+1) dimensional KMN equation are acquired. In particular, the one-soliton solution, two-soliton solution and three-soliton solution of the (2+1) dimensional KMN equation are given in detail. Substituting these solutions into Eq. (2) reveals that they satisfied Eq. (2) and are analyzed graphically. It can be seen from the picture that when \(\lambda \) $\lambda$ is purely imaginary, the interaction between solitons is periodic.