Complex excitations for the derivative nonlinear Schr\"{o}dinger equation

TL;DR

This paper develops concise Darboux transformation formulas for the derivative nonlinear Schrödinger equation, enabling the construction of complex excitations like multi-solitons, periodic solutions, and hybrid patterns, revealing new wave interaction phenomena.

Contribution

The paper introduces general semi-degenerate Darboux transformations for the DNLS equation, allowing systematic construction of diverse complex wave solutions and analyzing their properties.

Findings

Maximum amplitudes depend on spectral parameter sums

Interactions produce peaks with varying amplitudes and sizes

Solitons on periodic backgrounds behave like breathers

Abstract

The Darboux transformation (DT) formulae for the derivative nonlinear Schr\"{o}dinger (DNLS) equation are expressed in concise forms, from which the multi-solitons, n-periodic solutions, higher-order hybrid-pattern solitons and some mixed solutions are obtained. These complex excitations can be constructed thanks to more general semi-degenerate DTs. Even the non-degenerate N-fold DT with a zero seed can generate complicated n-periodic solutions. It is proved that the solution q[N] at the origin depends only on the summation of the spectral parameters. We find the maximum amplitudes of several classes of the wave solutions are determined by the summation. Many interesting phenomena are discovered from these new solutions. For instance, the interactions between n-periodic waves produce peaks with different amplitudes and sizes; A soliton on a single periodic wave background shares a similar feature as a breather due to the interference of the periodic background. In addition, the results are extended to the reverse-space-time DNLS equation.