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 $p^{\prime }\left(v\right)<0$ is a sufficient condition for instability, while $p^{\prime }\left(v\right)>0$ 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 $p\left(v\right)$ was calculated approximately by a collective coordinate theory, and the criterion was confirmed by simulations. The goal of this paper is to calculate $p\left(v\right)$ 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 $p\left(v\right)$ curve is calculated either analytically or numerically, and the stability criterion is confirmed.

Figure 1

Simulations of the logarithmic NLSE with a harmonic potential well. Left and right panels: $\omega =0.05,\lambda =1$ and $\omega =5,\lambda =1$, respectively. Upper panels: soliton profiles for different times: $t^{*}=0$ (solid line), $t^{*}=T/4$ (dotted line), and $t^{*}=T/2$ (dashed line). Lower panels: soliton width $\sigma \left(t\right)$ oscillates with frequency $1.4078$ (left panel) and $10.53$ (right panel).

Figure 2

Profile $|u(x,t^{*}){|}^2$ of a bisoliton in a parabolic potential well for different times in units of $T=\pi$ (solid line $t^{*}=0$, dotted line $t^{*}=2T$, dashed line $t^{*}=6T$). Simulations of Eq. (51) with the potential (52), with ${\lambda }_0={\mu }_0=0,\mu =1$, and $\lambda =2$. IC: Eq. (53) with $\eta \left(0\right)=1,\xi \left(0\right)=0,c=1$.

Figure 3

Position $q\left(t\right)$ and momentum $P\left(t\right)$ of the right soliton of the bisoliton in Fig. 2. The slope of the $p\left(v\right)$ curve is $0.992$ with a correlation coefficient $0.99998$.

Figure 4

Stability curves for parametrically driven solitons. IC: Eq. (55) with ${\eta }_0=0.25$ (upper left and right panels without and with the use of fitting functions, respectively), ${\eta }_0=0.35$ (lower left panel), ${\eta }_0=0.48$ (lower right panel). Other ICs: $q_0=0,{\Phi }_0=\pi /2,p_0=-K/4$. Parameters: $r=0.05,K=-0.1,\delta =-1$.