Application of Riemann-Hilbert method to an extended coupled nonlinear Schrodinger equations

Based on the non-semisimple Lie algebra (g) over tilde, we introduce a new integrable coupling system of the focusing nonlinear Schrodinger equation associated with a 4th-order block matrix spectral problem, which is called the extended coupled nonlinear Schrodinger (ECNLS) equations. The ECNLS equations can be reduced to many equations with physical background, including the focusing nonlinear Schrodinger equations (NLS) equation, the new integrable system of coupled NLS equations, the Manakov system, the mixed coupled NLS equations, etc. Based on the 4th-order spectral problem, we analyze the asymptotic behavior, analyticity and symmetry of the eigenfunctions and scattering coefficients. It follows that a formulation of solutions is developed for the Riemann-Hilbert problems associated with the reflectionless transforms. Furthermore, the N-soliton solutions of the ECNLS equations are generated. Additionally, the ECNLS equations are extended to a multi-component nonlinear Schrodinger system by means of a new N x N non-semisimple Lie algebra (g) over tilde, which means that the ECNLS equations are extended to an arbitrary number of components. Actually, the coupled and multi-component equations that we obtained can enrich the existing integrable models and possibly describe new nonlinear phenomena. (C) 2022 Elsevier B.V. All rights reserved.

Automated summary

In this paper, a new integrable coupling system of the focusing nonlinear Schrodinger equation, called the extended coupled nonlinear Schrodinger (ECNLS) equations, is introduced based on the non-semisimple Lie algebra. The asymptotic behavior, analyticity, and symmetry of the eigenfunctions and scattering coefficients are analyzed based on a 4th-order block matrix spectral problem. Solutions are formulated for the Riemann-Hilbert problems associated with the reflectionless transforms, and N-soliton solutions of the ECNLS equations are generated. Furthermore, the ECNLS equations are extended to a multi-component nonlinear Schrodinger system, allowing for an arbitrary number of components and the possibility of describing new nonlinear phenomena.