The Riemann-Hilbert Approach to Initial-Boundary Value Problems for Integrable Coherently Coupled Nonlinear Schrödinger Systems on the Half-Line

Abstract

An integrable coherently coupled nonlinear Schrödinger system describing the propagation of polarised optical waves in an isotropic medium with a generalized 4 × 4 matrix Ablowitz-Kaup-Newell-Segur-type Lax pair is studied. The corresponding initial-boundary value problem is reduced to a matrix Riemann-Hilbert problem in the complex plane. Moreover, it is shown that the associated spectral functions depend on each other and satisfy a global relationship.

Riemann-Hilbert method and N-soliton solutions for the mixed Chen-Lee-Liu derivative nonlinear Schrödinger equation

“…Since the RH method was proposed, many solutions of soliton equations have been discussed, such as, the coupled DNLS equation [16], the coupled higher-order NLS equation [17], the short pulse(SP) equation [18], the coupled modified Korteweg-de Vries (mKdV) equation [19], the generalized Sasa-Satsuma equation [20], the twocomponent Gerdjikov-Ivanov(GI) equation [21], the modified SP equation [22], and so on [23,24,25]. In particular, RH method is an effective way to working the initial-boundary value problems(IBVPs) of the integrable systems [26,27,28,29,30,31].…”

Riemann-Hilbert approach for a mixed coupled nonlinear Schrödinger system and its soliton solutions

“…The main idea of the this method is to established a corresponding matrix RH problem on the Lax pair of integrable equations. Furthermore, the RH method is also an effective way to examined the initial boundary value problems [29,30,31,32,33,34,35] and the long-time asymptotic behavior [36] of the integrable nonlinear evolution PDEs.…”