Double-flat-top half-vortices and self-bound solitary wave billiards in cubic-quintic media with intermodal attraction

TL;DR

This work analyzes two-component beams in a cubic-quintic (CQ) nonlinear medium with cross-phase modulation, showing that intermodal attraction can yield stable double-flat-top half-vortex solitons where the components carry different topological charges ($m_2=0$, $m_1 eq0$). Using an asymptotic expansion near the second component's flat background and numerical continuation with ${eta}=2$, ${ extalpha}=0$, the authors derive an effective 2D cubic NLS for the first component and demonstrate the formation of a second plateau atop the background, yielding double-flat-top states. Dynamical simulations reveal that unstable vortices split into fragments that interact with the flat-top boundary in a self-bound billiard-like manner, with speculative chaotic behavior and energy exchange during collisions. The results connect to the liquid-light analogy and suggest robust, self-contained confinement mechanisms with potential applicability to optical, atomic, and plasma contexts, and may be approachable via variational methods as well.

Abstract

We consider a bimodal light field envelope propagating in a bulk medium characterized by competing cubic and quintic nonlinearities. The subfields are coupled by a cross-phase modulation term and experience effective attraction. We find dynamically stable stationary states which have two distinct flat-top regions with different intensities. These solutions represent half-vortices, where the first and second components are essentially different and, in particular, carry different topological charges: zero for one component and nonzero for the other. The typical propagation of an unstable half-vortex leads to the splitting of the central vortex core into several fragments which quasielastically interact with the boundary of the flat-top region. This behavior is interpreted as a self-bound solitary wave billiard, where the emerging fragments are the billiard balls and the flat-top region is the dynamically deforming table.

Figures (7)

• Figure 1: Radial profiles of stationary states with zero ($m_1=0$) and single ($m_1=1$) topological charge in the first component for three different propagation constants $b_1$ in the first component and fixed propagation constant in the second component, $b_2 = 0.1835$ (which is slightly below the cutoff propagation constant $\tilde{b}_2 = 3/16\approx 0.1875$). The lowest panel illustrates the double-flat-top solutions. The topological charge of the second component is always zero: $m_2=0$.
• Figure 2: (a) Dependencies of the energy flow in the first component, $P_1$, on the propagation constant in the same component, $b_1$, for several values of the topological charge, $m_1=0,1,2,3$, while the propagation constant in the second component, $b_2$, is fixed to the same value as in Fig. \ref{['fig:profiles']}. Fragments with thin and thick lines represent unstable and stable solutions, respectively. To improve the clarity, we have omitted the interval $b_1 \in (1.6, 3)$ from the horizontal axis. All solutions in the omitted range are unstable, except for those with $m_1=0$. The dashed ellipse with the label "asymptotic limit" highlights the region where the solutions are born in the asymptotic limit, i.e., as $\varepsilon \to 0$. Panel (b) shows a closer view of the same dependencies, focusing on the region where the double-flat-top branches meet with the single-flat-top ones.
• Figure 3: Examples of stable propagation of double-flat-top states with vorticity $m_1=1$ (a,b) and $m_1 = 2$ (c,d). Large pseudocolor plots show the amplitude distribution, $|\psi_2|$, in the second component at $z=0$ (a,c) and $z=2\times 10^3$ (b,d). The insets show the pseudocolor plots of the phase distribution, $\varphi = \arg \psi_1$, in the first component, where black and white colors correspond to $\varphi=-\pi$ and $\varphi=\pi$, respectively. Large plots show the amplitude distributions within the window $(x,y) \in [-50, 50]\times [-50,50]$, and the insets show the phase distributions within the window $(x,y) \in [-15, 15]\times [-15,15]$. The shown solutions correspond to the propagation constant $b_1 = 3.3674$.
• Figure 4: Propagation of an unstable state with the unitary topological charge in the first component, i.e., $m_1=1$ and with $b_1 = 3.03$. The snapshots taken at different propagation distances $z$ show the amplitude of the second component, $|\psi_2|$, within the window $(x,y) \in [-45, 45]\times [-45, 45]$. The lines with arrows show the fragments of trajectories along which the emerged fragments move, and the direction of the arrows indicates an increase in the propagation distance $z$. The rightmost plot shows the amplitude distribution at the propagation distance past the irreversible coalescence of two fragments into a single one. In this plot, the complete trajectories of both fragments are shown, starting from the propagation distance where they have emerged from the vortex splitting instability and ending at the propagation distance where they have coalesced. The amplitude of the first component, $|\psi_1|$, behaves in a similar way, but the high-intensity spots move on a zero background, i.e., there is no flat-top plateau in the first component. See Supplemental Material for the multimedia file corresponding to this simulation.
• Figure 5: Examples of unstable propagation for states with double ($m_1=2$, a--c) and triple ($m_1=3$, d--f) topological charge in the first component. The snapshots taken at different propagation distances $z$ show the amplitude of the second component, $|\psi_2|$, within the window $(x,y) \in [-45, 45]\times [-45, 45]$. The lines with arrows show the fragments of trajectories along which the emerged fragments move, and the direction of the arrows indicates an increase in the propagation distance $z$. The propagation constants corresponding to the first components are equal to $b_1 \approx 3.01$ (a--c) and $b_1 \approx 3.03$ (d--f). See Supplemental Material for the multimedia files corresponding to these simulations.