A Riemann-Hilbert Approach to the Multicomponent Kaup-Newell Equation

Abstract

A Riemann-Hilbert approach is developed to the multicomponent Kaup-Newell equation. The formula is presented of N -soliton solutions through an identity jump matrix related to the inverse scattering problems with reflectionless potential.

Inverse Scattering Transform for the Generalized Derivative Nonlinear Schrödinger Equation via Matrix Riemann–Hilbert Problem

“…If det(P 1 ) ≠ 0 in their analytic domain, we can solve this Riemann-Hilbert problem by Plemelj formula. Under the canonical normalization equation (17) and equation (21), the solution to the regular Riemann-Hilbert problem is provided by the following integral equation:…”

“…Researchers found that it was simpler and more e ective than the original IST. On the one hand, scholars have used the RH approach to obtain the solutions to many integrable equations, such as the coupled derivative NLS (DNLS) equation [8], the coupled higher-order NLS equation [9], the generalized Camassa-Holm equation [10], the short pulse (SP) equation [11], and other integrable equations [12][13][14][15][16][17][18][19][20][21][22][23][24][25]. On the other hand, the RH method has been considered to be an e ective way to work the initialboundary value (IBV) problems of the integrable nonlinear evolution PDEs [26][27][28][29].…”

Dynamic behaviors of soliton solutions for a three-coupled Lakshmanan–Porsezian–Daniel model

“…These solutions can further help to understand natural phenomena and laws. In recent years, more and more mathematical physicists have begun to pay attention to Riemann-Hilbert (RH) approach [9,10], which is a new powerful method for solving integrable linear and nonlinear evolution equations [11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. The main idea of this method is to establish a corresponding matrix RH problem on the Lax pair of integrable equations.…”