Speed of wave packets and the nonlinear Schrödinger equation

The universal theory of weakly nonlinear wave packets given by the nonlinear Schrödinger equation is revisited. In the limit where the group and phase velocities are very close together, a multiple-scale analysis carried out beyond all orders reveals that a single soliton, bright or dark, can travel at a different speed than the group velocity. In an exponentially small but finite range of parameters, the envelope of the soliton is locked to the rapid oscillations of the carrier wave. Eventually, the dynamics is governed by an equation analogous to that of a pendulum, in which the center of mass of the soliton is subjected to a periodic potential. Consequently, the soliton speed is not constant and generally contains a periodic component. Furthermore, the interaction between two distant solitons can in principle be profoundly altered by the aforementioned effective periodic potential and we conjecture the existence of new bound states. These results are derived on a wide class of wave models and in such a general way that they are believed to be of universal validity.

Figure 1

Top: Phase portrait of Eq. (8) in the case $\eta =1$. The blue lines are the separatrices that joins the unstable stationary points (red crosses). Black dots are stable stationary points. Note that well above the separatrices, ${\dot{x}}_0$ becomes nearly constant, so that the pinning force exerted by the carrier wave on the envelope becomes negligible. Bottom: wave packets traveling exactly at the phase velocity and corresponding to stable (black) and unstable (red) configurations in an inversion-symmetric system ($\kappa =2$). In this illustration, $\varepsilon =0.2$.

Figure 2

Average soliton speed $\langle v\rangle$ in the vicinity of ${\omega }_0$. Inside the locking range $[{\omega }_0-\mathrm{\Delta }{\omega }_l/2,{\omega }_0+\mathrm{\Delta }{\omega }_l/2]$, the soliton envelope is locked to the phase velocity $v_p\left(\omega \right)$. Moving away from the locking range, $\langle v\rangle$ gradually tends to the group velocity $v_g\left(\omega \right)$. Inset: examples of soliton trajectories just inside (orange) and just outside to the right of the locking range (blue).

Figure 3

Locking range, ${\varepsilon }^{-1/2}exp-\kappa \pi /4\varepsilon$, as a function of the scale separation $\varepsilon$ and of the system inversion symmetry ($\kappa =1,2$.)