Transformation from the nonautonomous to standard NLS equations

Abstract

In this paper we show a systematical method to obtain exact solutions of the nonautonomous nonlinear Schrödinger (NLS) equation. An integrable condition is first obtained by the Painlevé analysis, which is shown to be consistent with that obtained by the Lax pair method. Under this condition, we present a general transformation, which can directly convert all allowed exact solutions of the standard NLS equation into the corresponding exact solutions of the nonautonomous NLS equation. The method is quite powerful since the standard NLS equation has been well studied in the past decades and its exact solutions are vast in the literature. The result provides an effective way to control the soliton dynamics. Finally, the fundamental bright and dark solitons are taken as examples to demonstrate its explicit applications.

Bright and dark solitons in a quasi-1D Bose–Einstein condensates modelled by 1D Gross–Pitaevskii equation with time-dependent parameters

“…Another interesting aspect of the temporal modulations is to look into the problem of parametric amplification, which however falls outside the scope of the present analysis and should be dealt with separately. It may also be noted that in the context of optical systems with varying coefficients, only picosecond optical solitons have been observed [59][60][61][62][63][64][65].…”

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.…”

“…The law of soliton adaptation to an external potential has come as a surprise and this law is being today the object of much concentrated attention in the field. The interested reader can find many important results and citations, for example, in the papers published recently by Zhao et al Luo et al, 2009;Zhao et al, 2009;2008), Shin (Shin, 2008) and (Kharif et al, 2009;Porsezian et al, 2007;Yan, 2010). How can we determine whether a given nonlinear evolution equation is integrable or not?…”

Influence of generalized external potentials on nonlinear tunneling of nonautonomous solitons: Soliton management

“…The transformation from a nonautonomous NLSE to a standard autonomous NLSE under some proper conditions is discussed in [10]. Under dispersion and nonlinearity management, both chirp free and chirped nonautonomous solitons have been investigated in [11].…”