Double and Triple-Pole Solutions for the Third-Order Flow Equation of the Kaup-Newell System with Zero/Nonzero Boundary Conditions

TL;DR

This paper develops explicit double and triple-pole soliton solutions for the third-order flow Kaup-Newell system under zero and non-zero boundary conditions using the Riemann-Hilbert approach, revealing effects of higher-order dispersion.

Contribution

It systematically derives N-double and N-triple pole solutions for the third-order Kaup-Newell system with detailed analysis, extending classical results to higher-order flows.

Findings

Double and triple-pole solutions are explicitly constructed.

Higher-order dispersion alters soliton trajectories and velocities.

Asymptotic behavior of solitons is analyzed for large time.

Abstract

In this work, the double and triple-pole solutions for the third-order flow equation of Kaup-Newell system (TOFKN) with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) are investigated by means of the Riemann-Hilbert (RH) approach stemming from the inverse scattering transformation. Starting from spectral problem of the TOFKN, the analyticity, symmetries, asymptotic behavior of the Jost function and scattering matrix, the matrix RH problem with ZBCs and NZBCs are constructed. Then the obtained RH problem with ZBCs and NZBCs can be solved in the case of scattering coefficients with double or triple zeros, and the reconstruction formula of potential, trace formula as well as theta condition are also derived correspondingly. Specifically, the general formulas of $N$-double and $N$-triple poles solutions with ZBCs and NZBCs are derived systematically by means of determinants. The vivid plots and dynamics analysis for double and triple-pole soliton solutions with the ZBCs as well as double and triple-pole interaction solutions with the NZBCs are exhibited in details. Compared with the most classical second-order flow Kaup-Newell system, we find the third-order dispersion and quintic nonlinear term of the Kaup-Newell system change the trajectory and velocity of solutions. Furthermore, the asymptotic states of the 1-double poles soliton solution and the 1-triple poles soliton solution are analyzed when $t$ tends to infinity.