Darboux Transformations, Higher-Order Rational Solitons and Rogue Wave Solutions for a (2 + 1)-Dimensional Nonlinear Schrödinger Equation**Supported by National Natural Science Foundation of China under Grant Nos. 11775121 and 11435005, and K.C. Wong Magna Fund in Ningbo University

Abstract

By Taylor expansion of Darboux matrix, a new generalized Darboux transformations (DTs) for a (2 + 1)-dimensional nonlinear Schrödinger (NLS) equation is derived, which can be reduced to two (1 + 1)-dimensional equation: a modified KdV equation and an NLS equation. With the help of symbolic computation, some higher-order rational solutions and rogue wave (RW) solutions are constructed by its (1, N −1)-fold DTs according to determinants. From the dynamic behavior of these rogue waves discussed under some selected parameters, we find that the RWs and solitons are demonstrated some interesting structures including the triangle, pentagon, heptagon profiles, etc. Furthermore, we find that the wave structure can be changed from the higher-order RWs into higher-order rational solitons by modulating the main free parameter. These results may give an explanation and prediction for the corresponding dynamical phenomena in some physically relevant systems.