The Darboux transformation of the derivative nonlinear Schr\"odinger equation

TL;DR

This paper develops explicit formulas for the Darboux transformation of the derivative nonlinear Schrödinger equation, enabling the construction of various solutions including new rogue wave and rational traveling wave solutions.

Contribution

It provides a novel explicit determinant-based formulation of the n-fold Darboux transformation for the DNLS equation, facilitating the generation of diverse solutions.

Findings

Explicit formulas for n-fold Darboux transformation elements.

Derivation of rogue wave and rational traveling solutions.

Complete classification of solutions generated by one-fold DT.

Abstract

The n-fold Darboux transformation (DT) is a 2\times2 matrix for the Kaup-Newell (KN) system. In this paper,each element of this matrix is expressed by a ratio of $(n+1)\times (n+1)$ determinant and $n\times n$ determinant of eigenfunctions. Using these formulae, the expressions of the $q^{[n]}$ and $r^{[n]}$ in KN system are generated by n-fold DT. Further, under the reduction condition, the rogue wave,rational traveling solution, dark soliton, bright soliton, breather solution, periodic solution of the derivative nonlinear Schr\"odinger(DNLS) equation are given explicitly by different seed solutions. In particular, the rogue wave and rational traveling solution are two kinds of new solutions. The complete classification of these solutions generated by one-fold DT is given in the table on page.