A refined empirical stability criterion for nonlinear Schroedinger solitons under spatiotemporal forcing

TL;DR

This work refines the stability criterion for driven cubic NLS solitons by leveraging a modified four-parameter collective-coordinate framework to compute a curve $p(v)$, whose negative-slope segments indicate instability. The approach remains valid across constant, harmonic, and biharmonic spatiotemporal forcings and in the presence or absence of damping, and it is corroborated by direct simulations of the full PDE. The authors also use phase-portrait analysis for constant $K$ to connect orbit sense with stability and demonstrate ratchet-like behavior under biharmonic driving, emphasizing the importance of using the normalized momentum $p$ rather than the canonical momentum $P$. The results advance practical stability diagnostics and provide a pathway to control soliton motion in forced NLS systems, with potential extensions to other nonlinearities.

Abstract

We investigate the dynamics of travelling oscillating solitons of the cubic NLS equation under an external spatiotemporal forcing of the form $f(x,t) = a \exp[iK(t)x]$. For the case of time-independent forcing a stability criterion for these solitons, which is based on a collective coordinate theory, was recently conjectured. We show that the proposed criterion has a limited applicability and present a refined criterion which is generally applicable, as confirmed by direct simulations. This includes more general situations where $K(t)$ is harmonic or biharmonic, with or without a damping term in the NLS equation. The refined criterion states that the soliton will be unstable if the "stability curve" $p(\v)$, where $p(t)$ and $\v(t)$ are the normalized momentum and the velocity of the soliton, has a section with a negative slope. Moreover, for the case of constant $K$ and zero damping we use the collective coordinate solutions to compute a "phase portrait" of the soliton where its dynamics is represented by two-dimensional projections of its trajectories in the four-dimensional space of collective coordinates. We conjecture, and confirm by simulations, that the soliton is unstable if a section of the resulting closed curve on the portrait has a negative sense of rotation.

Figures (8)

• Figure 1: Stability curve $p(\mathrm{v})$ corresponding to $a=0.05$, $K=0.1$, $\delta=-1$ and $\beta=0$. The initial conditions for the Eqs. (\ref{['be1']})-(\ref{['be4']}) were $q_{0}=p_{0}=\Phi_{0}=0$ and a) $\eta_{0}=0.8$, b) $\eta_{0}=0.65$, c) $\eta_{0}=0.1$. The integration time $t_{f}=1000$.
• Figure 2: The phase portrait of the system (\ref{['be1']})-(\ref{['be4']}) with $a$, $K$, $\delta$ and $\beta$ as in Fig. (\ref{['fig1']}). Shown is Im$\Psi(X= {\overline \mathrm{v}} t-V_f t,t)$ versus Re$\Psi(X= {\overline \mathrm{v}} t-V_f t,t)$. The large ellipse corresponds to $\eta_{0}=0.8$, the horseshoe to $\eta_{0}=0.65$ and the small ellipse to $\eta_{0}=0.1$. Other initial conditions are as in Fig. (\ref{['fig1']}). The separatrix is shown by the dotted curve. The filled and open circles are stable and unstable fixed points, respectively.
• Figure 3: Soliton amplitude $\eta(t)$ from the collective coordinates theory (solid lines) and from the simulations (dashed lines). The parameters and the initial conditions are the same as in Fig. \ref{['fig1']}. a) $\eta_{0}=0.8$, b) $\eta_{0}=0.65$, c) $\eta_{0}=0.1$ (shown are results for early times $0 \le t \le 100$); d) $\eta_{0}=0.1$ (shown are simulation results for late times $600 \le t \le 1000$.)
• Figure 4: Stability diagram near $\eta_{c}^{(1)} = 0.684$, with $q_{0} = \Phi_{0} = 0$. Parameters: $a = 0.05$, $K =0.1$, $\delta = -1$, $\beta = 0$. Circles: unstable soliton. Plus: stable soliton. Dashed line: for $p_0=K$, the $p(\mathrm{v})$ curve is a point.
• Figure 5: Collective coordinates results for harmonic $K(t)$, no damping. $a=0.05$, $k=-0.1$, $\omega=0.02$, $\theta=0$, $\delta=-3$, $\beta=0$, $q_{0}=p_{0}=0$, $\Phi_{0}=\pi/2$, $\eta_{0}=1$. a) $q(t)$ exhibits $\omega_{i}$-oscillations modulated by the frequency $\omega$. Shown is the interval $0 \le t \le T_{d}= 2 \pi/ \omega$. b) $q(t)$ exhibits $\omega$-oscillations modulated by the frequency $\omega_{l}=2 \pi/T_{l}$. Here $0 \le t \le 8000$. c) Stability curve $p(\mathrm{v})$ for $T_{d}/2 -5 \le t \le T_{d}/2 +10$. The arrow points to the section of the curve with a negative slope loop.