Vortex revivals and Fermi-Pasta-Ulam-Tsingou recurrence

We study the self-trapped vortex-ring eigenstates of the two-dimensional Schrödinger equation with focusing Poisson and cubic nonlinearities. For each value of the topological charge $l$, there is a family of solutions depending on a parameter that can be understood as the relative importance of the cubic term. We analyze the perturbative stability of the solutions and simulate the fate of the unstable ones. For $l=1$ and $l=2$, there is an interval of the family of eigenstates for which the initial profile breaks apart but subsequently reconstructs itself, a process that can be interpreted as a nontrivial nonlinear oscillation between the vortex and an azimuthon. This revival provides a compelling realization of a recurrence of the Fermi-Pasta-Ulam-Tsingou type. Outside this interval, the vortices can be stable (for small cubic terms) or unstable and nonrecurrent (for large cubic terms). We argue that there is a crossover between these regimes that resembles a strong stochasticity threshold. For $l\ge 3$, all solutions are unstable and nonrecurrent. Finally, we comment on the possible experimental implementation of this phenomenon in the context of nonlinear laser beam propagation in thermo-optical media.

Figure 1

Vortex-ring eigenstates. (a) Some eigenstates for $l=0,\cdots ,4$. (b) The power $P$ in terms of the parameter $f_0$. [(c)–(e)] Re($\delta$) as a function of $P$ for $l=1$, 2, 3, respectively. The integer associated with each line is the order of the azimuthal perturbation $p$. The solid line corresponds to $p=1$, the dotted line to $p=2$, the dashed line to $p=3$, and the dot-dashed line to $p=4$. The insets show more detailed views of the transition regions from stability [no solution with $\mathrm{Re}\left(\delta \right)>0$] to perturbative instability. For $l\ge 3$, all solutions are unstable.

Figure 2

An unstable nonrecurrent vortex. Top: An example ($l=1$, $f_0=0.9$) of the long-distance evolution of $\sigma \left(z\right)$. Bottom: The two-dimensional profile of $|{\psi }_{\mathrm{vort}}{|}^2$ at different values of $z$. In region A there are partial revivals. Region B can be interpreted as the two pieces orbiting around each other (rotation is counter-clockwise), a behavior that becomes destabilized in region C. In (D), an $l=0$ soliton remains, but surrounded by some radiation.

Figure 3

A recurrent vortex. Top: An example ($l=1$, $f_0=0.325$) of long-distance evolution of $\sigma \left(z\right)$. Bottom: The two-dimensional profile of $|{\psi }_{\mathrm{vort}}{|}^2$ for several values of $z$ within the region A, indicated in the $\sigma \left(z\right)$ plot. The profiles show how the instability begins and breaks the beam into two pieces and how the initial state self-assembles.

Figure 4

A recurrent vortex and its eventual disintegration. Top: An example ($l=2$, $f_0=0.62$) of long-distance evolution of $\sigma \left(z\right)$. Bottom: The $|{\psi }_{\mathrm{vort}}{|}^2$ profile for several values of $z$.

Figure 5

Comparison of different qualitative behaviors. The evolution of $\sigma \left(z\right)$ with different values of $f_0$, for $l=1$ (a) and $l=2$ (b).

Figure 6

Timescales. The frequency of revivals ${\nu }_s$ (numerically computed points are represented by circles) and the distance before the onset of chaos $Z_c$ (numerically computed points are represented by diamonds, with those placed at $Z_c=3500$, meaning that $Z_c$ is larger than that value) as a function of $f_0$. Increasing $f_0$ corresponds to increasing the relative importance of the local cubic nonlinearity. The upper panel corresponds to the $l=1$ family of solutions and the lower panel to $l=2$.