Riemann-Hilbert approach and N-soliton formula for the N-component Fokas-Lenells equations

TL;DR

This paper develops a Riemann-Hilbert method to derive N-soliton solutions for the generalized N-component Fokas-Lenells equations, extending previous work from N=2 to arbitrary N and analyzing their soliton interactions.

Contribution

It introduces a Riemann-Hilbert framework for solving the N-component Fokas-Lenells equations and derives explicit N-soliton formulas for any positive integer N.

Findings

Derived N-soliton solutions for N=2,3,4

Analyzed soliton interactions and dynamics

Extended previous N=2 results to general N

Abstract

In this work, the generalized $N$-component Fokas-Lenells(FL) equations, which have been studied by Guo and Ling (2012 J. Math. Phys. 53 (7) 073506) for $N=2$, are first investigated via Riemann-Hilbert(RH) approach. The main purpose of this is to study the soliton solutions of the coupled Fokas-Lenells(FL) equations for any positive integer $N$, which have more complex linear relationship than the analogues reported before. We first analyze the spectral analysis of the Lax pair associated with a $(N+1)\times (N+1)$ matrix spectral problem for the $N$-component FL equations. Then, a kind of RH problem is successfully formulated. By introducing the special conditions of irregularity and reflectionless case, the $N$-soliton solution formula of the equations are derived through solving the corresponding RH problem. Furthermore, take $N=2,3$ and $4$ for examples, the localized structures and dynamic propagation behavior of their soliton solutions and their interactions are discussed by some graphical analysis.