The general mixed nonlinear Schrödinger equation: Darboux transformation, rogue wave solutions, and modulation instability

Wenbo Li,Chunyan Xue,Lili Sun

doi:10.1186/s13662-016-0937-9

Abstract

In this paper, the Darboux transformation method has been successfully applied to a general mixed nonlinear Schrodinger equation and some rogue wave solutions are proposed. First of all, the determinant representation of an n-fold DT is given explicitly. Then starting with a periodic seed solution, we obtain some rogue wave solutions of the general mixed nonlinear Schrodinger equation through iteration of a generalized DT. Second, the three-dimensional images and density profiles of the rogue waves are plotted to show the structures of these rogue wave solutions. Finally, we give evidence for the connection between the occurrence of rogue wave solutions and the modulation instability.

Highlights

In the past decades, there are extensive advancements in the field of nonlinear integrable systems
The DT is well known to be a powerful method for finding exact solutions of integrable systems [ ]
It is well known that some exact solutions of the mixed nonlinear Schrödinger (NLS) equation ( ) with α = have been constructed via DT [ – ] and the Hirota method [, ]

Summary

Introduction

There are extensive advancements in the field of nonlinear integrable systems. It is well known that some exact solutions of the mixed NLS equation ( ) with α = have been constructed via DT [ – ] and the Hirota method [ , ]. The main aim of this paper is to construct the DT to derive the rogue wave solutions of mixed NLS equation ( ) by using the generalized Darboux transformation, analyze the rogue wave through their figures. Equations ( ) and ( ) give us the new solutions after a Darboux transformation, and we can use the results of ( ) and ( ) to get two-order and three-order rogue wave solutions. We will consider the condition of the degeneration case of the Darboux matrix T k to get the exact rogue wave solutions of equation ( ). We obtain the first-order rational solution qr[ a]tional, which means the one-order rogue wave solution is qr[ w]

G G H e it

Modulation instability

Conclusions