Guided vortex bullets

Abstract

By means of the variational method and numerical simulations, we demonstrate the existence of stable 3D nonlinear modes, viz. vortex ``bullets'', in the form of pulsed beams carrying orbital angular momentum, that can self-trap in a 2D waveguiding structure. Despite the attractive self-interaction, which is necessary for producing the bullets (bright solitons), and which readily leads to the collapse in the 3D setting as well as to spontaneous splitting of vortex modes, we find a critical value of the trapping depth securing the stabilization of the vortex bullets. We identify experimental conditions for the creation of these topological modes in the context of coherent optical and matter waves. Collisions between the bullets moving in the unconfined direction are found to be elastic. These findings contribute to the understanding of self-trapping in nonlinear multidimensional systems and suggest new possibilities for the stabilization and control of 3D topological solitons.

Figures (5)

• Figure 1: a) Hamiltonian $H$ vs. norm $N$ for several values of the potential strength $\gamma$, as predicted by the VA. b) The VA-predicted dependences $N$ on the transverse bullet's radius $R$, for the same set of values of $\gamma$.
• Figure 2: Norm $N$ of the numerically found stationary vortex solutions vs. the propagation constant $\beta$ for different values of the waveguide strength parameter $\gamma$. Dashed and continuous lines indicate VK-unstable and VK-stable families respectively. The black (dashed) curve displays, for comparison, the family of unstable free-space solutions with $\gamma =0$. Doted lines represent the solutions obtained with the variational method. The inset shows numerical (solid) and variational (dotted) radial shapes of the solutions for $\beta =0.25$ [designated by the vertical gray line in the main plot and with the colors corresponding to those of the $N(\beta )$ curves], along with the Gaussian trapping profile $-V/\gamma$ (red curve).
• Figure 3: A characteristic stable numerical solution of Eq. ( \ref{['eq:final']}), with $\gamma =4$, $\beta =0.25$ and $\ell =1$. Displayed are contour and mesh views of $|\psi |$, in a) and b), respectively. Panel c) shows the 3D isosurface, and d) plots the radial and axial (longitudinal) profiles of $|\psi |$ at $z=0$ and $\rho =\rho _{\max }$ (at $\rho$ realizing the maximum of $|\psi |$).
• Figure 4: The simulated perturbed evolution of two unstable solutions for a) $\gamma =2$, $\beta =0.25$ and b) $\gamma =4$, $\beta =2$, plotted in the $(x,y)$ plane. c) The same as in b), but in the $(z,y)$ plane. d) The evolution of the field's amplitude (normalized to its value at $t=0$) for different solutions, as indicated (both stable, for $\gamma =4$, $\beta =0.15$ and $\beta=0.25$ and unstable for all other cases).
• Figure 5: The elastic head-on collision of two stable vortex bullets from Fig. \ref{['fig:fig2']} with opposite signs, which are set in motion along the $z$ axis with opposite initial velocities. The collision leads to the elastic rebound of the solitons. The frames a) to f) correspond, respectively, to evolution times $t=0.0$, $7.6$, $15.2$, $22.9,30.5,$ and $38.1$.