Analytical spatiotemporal soliton solutions to (3+1)-dimensional cubic-quintic nonlinear Schrödinger equation with distributed coefficients

Abstract

We obtain exact spatiotemporal periodic traveling wave solutions to the generalized (3+1)-dimensional cubic-quintic nonlinear Schrödinger equation with spatial distributed coefficients. For restrictive parameters, these periodic wave solutions acquire the form of localized spatial solitons. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and gain (or loss). We then demonstrate the nonlinear tunneling effects and controllable compression technique of three-dimensional bright and dark solitons when they pass unchanged through the potential barriers and wells affected by special choices of the diffraction and/or the nonlinearity parameters. Direct numerical simulation has been performed to show the stable propagation of bright soliton with 5% white noise perturbation.

Exact traveling wave solutions of some nonlinear evolution equations

“…We concisely show what is F-expansion method and how to use it to find various periodic wave solutions to nonlinear wave equations [14]. In this method a nonlinear partial differential equation (PDE)…”

“…Exact solutions to nonlinear partial differential equations play an important role in nonlinear physical science since they can provide much physical information and more insight into the physical aspects of the problem and thus lead to further applications. In recent years, many methods for obtaining explicit traveling and solitary wave solutions of NLEEs have been proposed such as inverse scattering transform method [2], Darboux transformation method [3,4], Hirota's bilinear method [5], Bäcklund transformation method [6], homogeneous balance method [7], solitary wave ansatz method [8,9], Jacobi elliptic function expansion method [10], the tanh function method [11], ð G 0 G Þ expansion method [12,13], F-expansion method [14], projective Ricatti equation method [15,16,17] and so on. Among them extended F-expansion and projective Ricatti equation methods have been proved to be a powerful mathematical tool to investigate the exact solutions for NLEEs.…”

Topological and non-topological soliton solution of the 1 + 3 dimensional Gross-Pitaevskii equation with quadratic potential term

“…When the coe cients are constant, the behavior of solutions toward the GPE strongly depends on the dimensionality of the problem. It is known that this equation supports solitons that are studied in the context of BEC and nonlinear optics [19][20][21]. Now, Eq.…”

“…(19) and (20) gives the expressions for free parameter, A, wave number, !, and dark soliton velocity, v. It is to be noted that the coe cients of the linearly independent functions tanh p 2 () in Eq. (19) are spontaneously zero for p = 1.…”

“…(19), equating the exponents p + 2 and 3p leads to p = 1. Setting coe cients of linearly independent functions tanh p+j () to zero for j = 0, 1, 2 in Eqs.…”