Dynamics of high-order solitons in the nonlocal nonlinear Schr\"{o}dinger equations

TL;DR

This paper investigates the complex dynamics of high-order solitons in three nonlocal nonlinear Schrödinger equations, revealing their diverse trajectories, structures, and interactions through a unified Riemann-Hilbert framework.

Contribution

It derives general high-order soliton solutions for three nonlocal NLS equations from a common Riemann-Hilbert approach, highlighting their unique symmetry relations and dynamic behaviors.

Findings

High-order fundamental solitons move on multiple trajectories with nearly equal velocities.

High-order multi-solitons exhibit complex wave structures and behaviors.

Hybrid-pattern solitons demonstrate nonlinear interactions between different soliton types.

Abstract

A study of high-order solitons in three nonlocal nonlinear Schr\"{o}dinger equations is presented, which includes the \PT-symmetric, reverse-time, and reverse-space-time nonlocal nonlinear Schr\"{o}dinger equations. General high-order solitons in three different equations are derived from the same Riemann-Hilbert solutions of the AKNS hierarchy, except for the difference in the corresponding symmetry relations on the "perturbed" scattering data. Dynamics of general high-order solitons in these equations is further analyzed. It is shown that the high-order fundamental-soliton is always moving on several different trajectories in nearly equal velocities, and they can be nonsingular or repeatedly collapsing, depending on the choices of the parameters. It is also shown that high-order multi-solitons could have more complicated wave structures and behave very differently from high-order fundamental solitons. More interesting is the high-order hybrid-pattern solitons, which are derived from combination of different size of block matrix in the Riemann-Hilbert solutions and thus they can describe a nonlinear interaction between several types of solitons.