A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup-Newell equation

TL;DR

This paper introduces a new, simpler form of soliton solutions for a higher-order Kaup-Newell equation, including multiple zeros solutions, using inverse scattering and dressing methods, with detailed dynamic analysis.

Contribution

It presents a novel, more direct method for constructing soliton solutions of a higher-order KN equation, including multiple zeros cases, improving upon previous approaches.

Findings

Derived new explicit soliton solutions

Constructed solutions with multiple zeros

Provided detailed dynamic visualizations

Abstract

Due to higher-order Kaup-Newell (KN) system has more complex and diverse solutions than classical second-order flow KN system, the research on it has attracted more and more attention. In this paper, we consider a higher-order KN equation with third order dispersion and quintic nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann-Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given, which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single soliton solution and two soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single-double zeros and double-double zeros are especially shown.