Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

A stability criterion for solitons of the driven nonlinear Schrödinger equation (NLSE) has been conjectured. The criterion states that p (v) < 0 is a sufficient condition for instability, while p (v) > 0 is a necessary condition for stability; here, v is the soliton velocity and p = P /N, where P and N are the soliton momentum and norm, respectively. To date, the curve p(v) was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations. The goal of this paper is to calculate p(v) exactly for several classes and cases of the generalized NLSE: a soliton moving in a real potential, in particular a time-dependent ramp potential, and a time-dependent confining quadratic potential, where the nonlinearity in the NLSE also has a time-dependent coefficient. Moreover, we investigate a logarithmic and a cubic NLSE with a time-independent quadratic potential well. In the latter case, there is a bisoliton solution that consists of two solitons with asymmetric shapes, forming a bound state in which the shapes and the separation distance oscillate. Finally, we consider a cubic NLSE with parametric driving. In all cases, the p(v) curve is calculated either analytically or numerically, and the stability criterion is confirmed.

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“…(33) can be converted by a modified Lens-type transformation, or a self-similar transformation [20][21][22], to the standard NLSE,…”

Section: -3mentioning

confidence: 99%

Soliton stability criterion for generalized nonlinear Schrödinger equations

2015

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“…In this paper, we first review a generalized pseudoconformal transformation introduced in [26] (lens transform in optics [27] see also [28]). As the first main result, we will use this generalized lens transformation to construct solutions of the general variable coefficient nonlinear Schrödinger equation (VCNLS):…”

Section: Introductionmentioning

confidence: 99%

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

et al. 2016

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Abstract:In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, dark-and Peregrine-type soliton solutions for NLS with variable coefficients. As an important application of solutions for the Riccati equation with parameters, by means of computer algebra systems, it is shown that the parameters change the dynamics of the solutions. Finally, we test numerical approximations for the inhomogeneous paraxial wave equation by the Crank-Nicolson scheme with analytical solutions found using Riccati systems. These solutions include oscillating laser beams and Laguerre and Gaussian beams.

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“…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.…”

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confidence: 99%

Anderson localization in the quintic nonlinear Schrödinger equation

2016

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In the present paper we consider the quintic defocusing nonlinear Schrödinger equation in presence of a disordered random potential and we analyze the effects of the quintic nonlinearity on the Anderson localization of the solution. The main result shows that Anderson localization requires a cutoff on the value of the parameter which controls the quintic nonlinearity, with the cutoff depending on the amplitude of the random potential.PACS numbers: 42.25.Dd, 42.65.Tg

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Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients

Cited by 95 publications

References 21 publications

Soliton stability criterion for generalized nonlinear Schrödinger equations

Soliton stability criterion for generalized nonlinear Schrödinger equations

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

Anderson localization in the quintic nonlinear Schrödinger equation

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