Solitons with cubic and quintic nonlinearities modulated in space and time

Abstract

This work deals with soliton solutions of the nonlinear Schrödinger equation with cubic and quintic nonlinearities. We extend the procedure put forward in a recent paper [J. Belmonte-Beitia, Phys. Rev. Lett. 100, 164102 (2008)], and we solve the equation in the presence of a linear background and cubic and quintic interactions which are modulated in space and time. As a result, we show how a simple parameter can be used to generate brightlike or darklike localized nonlinear waves which oscillate in several distinct ways, driven by the space and time dependence of the parameters that control the trapping potential and the cubic and quintic nonlinearities.

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics

“…It is now generally accepted that solitary waves in nonautonomous nonlinear and dispersive systems can propagate in the form of so-called nonautonomous solitons or solitonlike similaritons (see (Atre et al, 2006;Avelar et al, 2009;Belić et al, 2008;Chen et al, 2007;Hao, 2008;He et al, 2009;Hernandez et al, 2005;Hernandez-Tenorio et al, 2007;Liu et al, 2008;Porsezian et al, 2009;Serkin et al, 2007;Shin, 2008;Tenorio et al, 2005;Wang et al, 2008;Wu, Zhang, Li, Finot & Porsezian, 2008;Zhang et al, 2008;Zhao et al, 2009;2008) and references therein). Nonautonomous solitons interact elastically and generally move with varying amplitudes, speeds and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations.…”

Anderson localization in the quintic nonlinear Schrödinger equation

“…From the theoretical point of view, the control of such solutions can be facilitated through the search for analytical solutions of the 1D Gross-Pitaevskii equation (GPE). In this sense, recently, analytical solitonic solutions to the more general case, employing space-and time-dependent coefficients, was considered for the cubic [19], the cubic-quintic [20], the quintic [21], and also the GPE in higher dimensions [22]. Analytical breather solutions has been found in Ref.…”

Wide vector solitons in systems with time- and space-modulated nonlinearities

“…There are a vast variety of methods to obtain exact analytical solutions of the NLSE in 1 + 1 dimensions, such as the inverse scattering transform [3], the Hirota method [4], the similarity transformation method, which was applied for the first time to solve the nonautonomous CQSNLE in [5] and later in [6,7], and the point canonical transformations [8], which explains the origin of the ansatz involved in the similarity transformation method. The best-known solutions of the NLSE are those * luisarroyo@feg.unesp.br † dutra@feg.unesp.br ‡ marcelo.hott@pq.cnpq.br for solitary waves or solitons [9].…”