The Riemann–Hilbert approach for the Chen–Lee–Liu equation with higher-order poles

Abstract

This paper considers the Riemann–Hilbert problem (RHP) with higher-order poles for the Chen–Lee–Liu equation. By assuming the scattering coefficients have Nth order zero points, we obtain expressions of these scattering coefficients, the theta condition, and parallelization conditions between Jost solutions and their derivatives. Based on these parallelization conditions, it yields generalizations of residue conditions, which are essential for solving the RHP with higher-order poles. The explicit formulae for solutions of CLL equation are reconstructed from asymptotic behaviors of the RHP. As an example, we present the second-order pole soliton and illustrate that parameter b0 plays a principal role to determine the structures of solutions of the CLL equation.