Nonlinear Schrödinger Equation with Spatio-Temporal Perturbations

TL;DR

This work develops a collective-coordinate framework for the cubic nonlinear Schrödinger equation perturbed by a non-parametric spatio-temporal drive, damping, and a stabilizing linear term. By deriving ODEs for four CCs $(\eta,\xi,\zeta,\phi)$, it explains how a constant spatially periodic drive can induce net unidirectional soliton transport despite a zero-mean force, with small CC oscillations around a mean trajectory. The study identifies stability criteria via the slope of the momentum-velocity curve $P(V)$ and confirms them with NLSE simulations, while also deriving soliton and phonon dispersion relations from a canonical Hamiltonian formulation. These results yield a predictive framework for soliton stability and transport in perturbed NLSEs and lay groundwork for analyzing more complex driving protocols in future work.

Abstract

We investigate the dynamics of solitons of the cubic Nonlinear Schrödinger Equation (NLSE) with the following perturbations: non-parametric spatio-temporal driving of the form $f(x,t) = a \exp[i K(t) x]$, 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 ODEs for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force $f(x)$. 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 $f(x)$. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum $P(t)$ and soliton velocity $V(t)$: This is a parameter representation of a curve $P(V)$ 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.

Figures (8)

• Figure 1: Left panel: Soliton moving to the left for $t^{*}=250, 500$. Simulations of NLSE (solid lines) and numerical solutions of CC-equations (dotted lines). Right panel: Real (solid line) and imaginary (dashed line) parts of $u(x,t)$ for $t=500$. Parameters: $K=-0.1$, $a=0.05$, $\delta=-1$, $\beta=0.05$, with IC $\xi_{0}=0$, $\zeta_{0}=0$, $\phi_{0}=1.69$ 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 $\xi_{0}=0$, $\zeta_{0}=0$, $\phi_{0}=\pi/2$ 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 $\xi_{0}=0$, $\zeta_{0}=0$, $\phi_{0}=0$ and $\eta_{0}=\sqrt{K^{2}-\delta}/2$.
• Figure 4: The evolution of $\eta$, $\zeta$, $\xi$ and $\phi$ obtained from a simulation of the NLSE (dashed lines) and from the numerical solutions of the CC equations (solid lines). Parameters: $K=0.1$, $a=0.05$, $\delta=-1$, $\beta=0$, with IC $\xi_{0}=0$, $\zeta_{0}=0$, $\phi_{0}=\pi/2$ 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. \ref{['fig3']}.