Riemann-Hilbert problem for the nonlinear Schrödinger equation with multiple high-order poles under nonzero boundary conditions

Abstract

The Riemann-Hilbert (RH) problem is first developed to study the focusing nonlinear Schrödinger (NLS) equation with multiple high-order poles under nonzero boundary conditions. Laurent expansion and Taylor series are employed to replace the residues at the simple- and the second-poles. Further, the solution of RH problem is transformed into a closed system of algebraic equations, and the soliton solutions corresponding to the transmission coefficient $1/s_{11}(z)$ with an $N$-order pole are obtained by solving the algebraic system. Then, in a more general case, the transmission coefficient with multiple high-order poles is studied, and the corresponding solutions are obtained. In addition, for high-order pole, the propagation behavior of the soliton solution corresponding to a third-order pole is given as example.

19 pages, 5 figures

References in corpus (1)

• The Sasa-Satsuma equation with non-vanishing boundary conditions

Cited by in corpus (1)

• $N$-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithm