Two kinds of rogue waves of the general nonlinear Schr\"odinger equation with derivative

TL;DR

This paper demonstrates the designable integrability of a variable coefficient derivative nonlinear Schrödinger equation (VCDNLSE) and constructs rogue wave solutions with controllable profiles via explicit transformations from the standard DNLSE.

Contribution

It introduces a method to design coefficients of VCDNLSE analytically and derives two types of rogue wave solutions with different boundary conditions.

Findings

Two kinds of rogue waves with different boundary conditions are obtained.

The coefficients of VCDNLSE can be designed analytically using the transformation.

The integrability of VCDNLSE allows for potential control of rogue wave profiles in experiments.

Abstract

In this letter,the designable integrability(DI) of the variable coefficient derivative nonlinear Schr\"odinger equation (VCDNLSE) is shown by construction of an explicit transformation which maps VCDNLSE to the usual derivative nonlinear Schr\"odinger equation(DNLSE). One novel feature of VCDNLSE with DI is that its coefficients can be designed artificially and analytically by using transformation. What is more, from the rogue wave and rational traveling solution of the DNLSE, we get two kinds of rogue waves of the VCDNLSE by this transformation. One kind of rogue wave has vanishing boundary condition, and the other non-vanishing boundary condition. The DI of the VCDNLSE also provides a possible way to control the profile of the rogue wave in physical experiments.