Soliton stability criterion for generalized nonlinear Schrödinger equations

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.Grant No. 1146358 STP from the Alexander von Humboldt-Stiftung, Germany, through Research Fellowship for Experienced Researchers SPAMICINN (Spain) through FIS2011-24540Projects No. FQM207, No. P11-FQM7276, and No. P09-FQM-4643 by Junta de Andalucia (Spain