Dynamics of three nonisospectral nonlinear Schrödinger equations

Abstract

Dynamics of three nonisospectral nonlinear Schrödinger equations (NNLSEs), following different time dependencies of the spectral parameter, are investigated. First, we discuss the gauge transformations between the standard nonlinear Schrödinger equation (NLSE) and its first two nonisospectral counterparts, for which we derive solutions and infinitely many conserved quantities. Then, exact solutions of the three NNLSEs are derived in double Wronskian terms. Moreover, we analyze the dynamics of the solitons in the presence of the nonisospectral effects by demonstrating how the shapes, velocities, and wave energies change in time. In particular, we obtain a rogue wave type of soliton solutions to the third NNLSE.

New dynamics of the classical and nonlocal Gross-Pitaevskii equation with a parabolic potential

“…Classical case of the above equation has been well studied in [32]. Note that the classical GP equation (2) admits bilinear form and N -soliton solutions.…”

Soliton molecules and dynamics of the smooth positon for the Gerdjikov–Ivanov equation*

“…In recent years, adopting different representations of a general solution through zero background "seed" solution of the corresponding equations to obtain solutions is an important method to obtain positons, which is different from rouge waves. [30][31][32] Also, smooth positons of other systems were studied, such as the complex modified KdV equation, the derivative nonlinear Schrödinger equation, and the Kundu-Eckhaus equation. [33][34][35] To our knowledge, the soliton molecules, the dynamic of positon solution and hybrid solution for Eq.…”

Stable soliton propagation in a coupled (2 + 1) dimensional Ginzburg–Landau system*

“…[34] In previous work, some researchers have successfully derived the analytical solutions of NLSEs. [35][36][37][38][39][40][41][42][43] They have analyzed how to improve the quality of communication and develop different kinds of optical devices by controlling the soliton propagation. With research getting further, since actual systems often have energy exchange with the outside world, people noticed that traditional integrable systems are not enough to describe the soliton phenomena in reality.…”