Soliton stability criterion for generalized nonlinear Schrödinger equations

Abstract

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.

Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials

“…A general approach for studying soliton dynamics has been discussed for real potentials in the work of Quintero, Mertens and Bishop [20] and also by Kominis [3] for complex potentials. Here we follow the approach of [20].…”

“…Here we follow the approach of [20]. We are interested in solitary wave solutions that approach zero exponentially at ±∞.…”

“…We will show in what follows that this instability either leads to the solitary wave then oscillating at another frequency, or blowing up or collapsing. The usefulness of this criterion for studying soliton stability in generalized NLSEs was investigated in detail in [20]. In Sec.…”

Nonlinear Dirac equation solitary waves under a spinor force with different components

“…For the forced nonlinear Schrödinger (NLS) equation when subject to an external force of the form f (x) = r exp(−iKx), the authors found [48][49][50] that intrinsic soliton oscillations are excited, i.e., the soliton amplitude, width, phase, momentum, and velocity all oscillate with the same frequency. This behavior was predicted by a collective coordinates theory and was confirmed by numerical simulations.…”

A Space‐Time Fully Decoupled Wavelet Galerkin Method for Solving Multidimensional Nonlinear Schrödinger Equations with Damping

“…The nonlinearities ( ) and ( ) which are real-valued smooth functions with respect to the density = | | 2 are determined by the specific application. Note that, in this study, the most popular nonlinear term ( ) = with constant [1][2][3][4][5][6][7][8][9][10] has been extended to the general form ( ) which can be chosen as arbitrary function as needed, such as ( ) = 0 , ( ) = 1 (1 − − ), ( ) = /(1 + ), and ( ) = ln(1 + ), where 0 , and 1 are real constants [26,36]. Moreover, the general damping term ( ) which is usually ignored in many studies [8-16, 18-25, 27-34] is also considered in GNLSE (1), because the damping effect may play an important role and should not be ignored in some physical processes [7,17,26,35,36], such as the inelastic collisions in Bose-Einstein condensation [7,26].…”