Tail-free self-accelerating solitons and vortices

Self-accelerating waves in conservative systems, which usually feature slowly decaying tails, such as Airy waves, have drawn great interest in studies of quantum and classical wave dynamics. They typically appear in linear media, while nonlinearities tend to deform and eventually destroy them. We demonstrate, by means of analytical and numerical methods, the existence of robust one- and two-dimensional (1D and 2D) self-accelerating tailless solitons and solitary vortices in a model of two-component Bose-Einstein condensates, dressed by a microwave (MW) field, whose magnetic component mediates long-range interaction between the matter-wave constituents, with the feedback of the matter waves on the MW field taken into account. In particular, self-accelerating 2D solitons may move along a curved trajectory in the coordinate plane. The system may also include the spin-orbit coupling between the components, leading to similar results for the self-acceleration. The effect persists if the contact cubic nonlinearity is included. A similar mechanism may generate 1D and 2D self-accelerating solitons in optical media with thermal nonlinearity.

Figure 1

(a) Counterpropagating Raman beams $L_1$ and $L_2$ generate the spin-orbit-coupled two-component BEC, as shown in Ref. [52]. (b) The microwave field couples states 0 and 1 by an effective long-range interaction [see Eqs. (1) and (4)].

Figure 2

Typical density profiles in the (a1), (b1), and (c1) coordinate space and (a2), (b2), and (c2) momentum space (produced by the Fourier transform of the coordinate profile) of (a) a regular (single-peak) soliton, (b) a stripe soliton, and (c) a soliton of the plane-wave type, so called because it is carried by the plane-wave phase, for the pseudo-spin-up (solid lines) and pseudo-spin-down (short-dashed lines) components. These solutions to Eq. (1), with $K\equiv 1$ and ${\beta }_{1,2}=0$, were obtained, by means of the imaginary-time integration, at $\mathrm{\Omega }=1.5,\mathrm{\Omega }=0.7$, and $\mathrm{\Omega }=0.7$, with the respective chemical potentials $\mu =-1.4779,-0.7300$, and $-0.7298$. The wave functions predicted by the VA (long-dashed lines), ${\psi }_{\uparrow }^{\text{(VA)}}\left(x\right)$, based on the ansatz defined by Eqs. (10) and (11), with VA-predicted chemical potentials $-1.4781,-0.7262$, and $-0.7260$, respectively, are plotted too for comparison with the numerical results in the coordinate space. In (c1), the approximate wave function ${\psi }_{\downarrow }^{\text{(app)}}$ for the pseudo-spin-down component, produced by the separately developed analytical approximation (13), is plotted by the yellow dash-dotted line, which overlaps with its numerical counterpart.

Figure 3

Shown on top is the evolution of the same plane-wave-type soliton as in Fig. 2 (i.e., in the absence of the contact interactions, ${\beta }_{1,2}=0)$, initiated by the addition of a small random perturbation to it. In this figure and similar ones displayed below, the evolution is displayed by means of the map of the total density of both matter-wave components in the $(x,t)$ plane. Shown on the bottom is the soliton's average position $\langle x\left(t\right)\rangle \equiv {\int }_{-\infty }^{+\infty }{\mathrm{\Psi }}^{†}\left(x\right)\mathrm{\Psi }\left(x\right)xdx$ and velocity $d\langle x\left(t\right)\rangle /dt$ as functions of time. The time dependence of the velocity helps to evaluate the soliton's acceleration (weak jitter in $d\langle x\rangle /dt$ is caused by the randomness of the initial perturbation).

Figure 4

Same as in Fig. 3, but in the presence of the self-repulsive contact interactions in Eq. (1) with ${\beta }_1=-5.0$ and ${\beta }_2=0$.

Figure 5

Same as in Fig. 3, but in the presence of the self-attractive contact interactions in Eq. (1) with ${\beta }_1=0.3$ and ${\beta }_2=0$.

Figure 6

Plot of $|{\mathrm{\Phi }}_{\downarrow }{|}^2={\left|{\mathrm{\Phi }}_{\uparrow }\right|}^2$ for a stable self-accelerating vortex soliton with $a_x=10,a_y=0,V_{x,y}=0,{\beta }_{1,2}=0$ [no contact interactions in Eq. (21)], and $\gamma =\pi$. The right panel shows the time evolution of coordinate $x$ of the vortex center.

Figure 7

Same as in Fig. 6, but in the presence of relatively strong contact self-attraction, represented by the coefficient ${\beta }_1+{\beta }_2=10$ in Eq. (21).