Nonlinear Schrödinger equation with spatiotemporal perturbations

We investigate the dynamics of solitons of the cubic nonlinear Schrödinger equation (NLSE) with the following perturbations: nonparametric spatiotemporal driving of the form $f\left(x,t\right)=a\text{exp}\left[iK\left(t\right)x\right]$, damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a collective-coordinate-theory which yields a set of ordinary differential equations (ODEs) for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force $f\left(x\right)$. The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of $f\left(x\right)$. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum $P\left(t\right)$ and the soliton velocity $V\left(t\right)$: This is a parameter representation of a curve $P\left(V\right)$ which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.

Figure 1

Left panel: soliton moving to the left for $t^{\ast }=250,500$. Simulations of NLSE (solid lines) and numerical solutions of CC equations (dotted lines). Right panel: real (solid line) and imaginary (dashed line) parts of $u\left(x,t\right)$ for $t=500$. Parameters: $K=-0.1$, $a=0.05$, $\delta =-1$, $\beta =0.05$, with IC ${\xi }_0=0$, ${\zeta }_0=0$, ${\varphi }_0=1.69$, and ${\eta }_0=0.5$.

Figure 2

(Color online) The amplitude and position of the soliton obtained from a simulation of the NLSE (red dashed lines) and from the numerical solution of the CC equations (solid lines). Parameters: $K=-0.01$, $a=0.05$, $\delta =-3$, $\beta =0.01$, with IC ${\xi }_0=0$, ${\zeta }_0=0$, ${\varphi }_0=\pi /2$, and ${\eta }_0=1$.

Figure 3

(Color online) The amplitude and position of the soliton obtained from a simulation of the NLSE (red dashed lines) and from the numerical solution of the CC equations (solid lines). Parameters: $K=0.1$, $a=0.05$, $\delta =-3$, $\beta =0.001$, with IC ${\xi }_0=0$, ${\zeta }_0=0$, ${\varphi }_0=0$ and ${\eta }_0=\sqrt{K^2-\delta }/2$.

Figure 4

The evolution of $\eta$, $\zeta$, $\xi$, and $\varphi$ obtained from the numerical solutions of the CC equations (solid lines) and the evolution of $\eta$ and $\zeta$ from a simulation of the NLSE (dashed lines). Parameters: $K=0.1$, $a=0.05$, $\delta =-1$, $\beta =0$, with IC ${\xi }_0=0$, ${\zeta }_0=0$, ${\varphi }_0=\pi /2$, and ${\eta }_0=0.75$.

Figure 5

“Stability curve” $P$ versus $V$. Left panel: ${\eta }_0=0.75$. Right panel ${\eta }_0=0.76$. Other parameters as in Fig. 4.

Figure 6

The evolution of $\eta$, $\zeta$, $\xi$, and $\varphi$ obtained from the numerical solutions of the CC equations (solid lines) and the evolution of $\eta$ and $\zeta$ from a simulation of the NLSE (dashed lines). Parameters: $K=0.1$, $a=0.05$, $\delta =-1$, $\beta =0$, with IC ${\xi }_0=0$, ${\zeta }_0=0$, ${\varphi }_0=\pi /2$, and ${\eta }_0=0.76$. The simulations were carried forward to a final time $t_f=1000$.

Figure 7

“Stability curve” close to the boundary of an unstable regime. Parameters: $K=0.1$, $a=0.05$, $\delta =-1$, $\beta =0$, with IC ${\xi }_0=0$, ${\zeta }_0=0$, ${\varphi }_0=\pi /2$, and ${\eta }_0=0.46$.

Figure 8

The evolution of $\eta$, $\zeta$, $\xi$, and $\varphi$ obtained from the numerical solutions of the CC equations (solid lines) and the evolution of $\eta$ and $\zeta$ from a simulation of the NLSE (dashed lines). Parameters as in Fig. 7.