Rogue waves in the generalized derivative nonlinear Schrodinger equations

TL;DR

This paper derives explicit rogue wave solutions for a family of generalized derivative nonlinear Schrödinger equations using an improved bilinear KP reduction method, revealing universal features and new wave patterns.

Contribution

It introduces a simplified parameterization for rogue waves in GDNLS equations and demonstrates their universal form across different equations within this family.

Findings

Rogue waves expressed by elementary Schur polynomials with a new parameterization.

Maximum peak amplitude at order N is 2N+1 times the background, independent of specific GDNLS.

Identified new rogue wave patterns and analyzed background wavenumber effects.

Abstract

General rogue waves are derived for the generalized derivative nonlinear Schrodinger (GDNLS) equations by a bilinear Kadomtsev-Petviashvili (KP) reduction method. These GDNLS equations contain the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation as special cases. In this bilinear framework, it is shown that rogue waves to all members of these equations are expressed by the same bilinear solution. Compared to previous bilinear KP reduction methods for rogue waves in other integrable equations, an important improvement in our current KP reduction procedure is a new parameterization of internal parameters in rogue waves. Under this new parameterization, the rogue wave expressions through elementary Schur polynomials are much simpler. In addition, the rogue wave with the highest peak amplitude at each order can be obtained by setting all those internal parameters to zero, and this maximum peak amplitude at order $N$ turns out to be $2N+1$ times the background amplitude, independent of the individual GDNLS equation and the background wavenumber. It is also reported that these GDNLS equations can be decomposed into two different bilinear systems which require different KP reductions, but the resulting rogue waves remain the same. Dynamics of rogue waves in the GDNLS equations is also analyzed. It is shown that the wavenumber of the constant background strongly affects the orientation and duration of the rogue wave. In addition, some new rogue patterns are presented.