Application of Riemann–Hilbert method to an extended coupled nonlinear Schrödinger equations

Abstract

Based on the non-semisimple Lie algebra g˜, we introduce a new integrable coupling system of the focusing nonlinear Schrödinger equation associated with a 4th-order block matrix spectral problem, which is called the extended coupled nonlinear Schrödinger (ECNLS) equations. The ECNLS equations can be reduced to many equations with physical background, including the focusing nonlinear Schrödinger 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 Schrödinger system by means of a new N×N non-semisimple Lie algebra ĝ, 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.