Phase-controlled elastic, inelastic, and coalescent collisions of two-dimensional flat-top solitons

TL;DR

The paper addresses collisions of two-dimensional flat-top solitons in the cubic--quintic nonlinear Schrödinger equation, showing that the relative phase at impact largely determines whether collisions are elastic, inelastic, or lead to coalescence. Using real-time simulations, kinetic-energy diagnostics, and extracted effective interaction potentials, the authors reveal phase- and separation-dependent windows of elasticity and characterize attraction or repulsion between solitons. They further connect inelastic coalescence to interfacial energetics via a Young--Laplace–type balance and demonstrate, through variational energy minimization, that merged states sit in a robust energetic minimum. Together, these results provide a coherent framework linking phase interference, dynamical forces, and energetic stability for 2D flat-top solitons with broad implications for nonlinear optics and ultracold gases.

Abstract

We investigate elastic, inelastic, and coalescent collisions between two-dimensional flat-top solitons supported by the cubic-quintic nonlinear Schrödinger equation. Numerical simulations reveal distinct collision regimes ranging from nearly elastic scattering to strongly inelastic interactions leading to long-lived merged states. We demonstrate that the transition between these regimes is primarily controlled by the relative phase of the solitons at the collision point, with out-of-phase collisions suppressing overlap and in-phase collisions promoting strong interaction. Kinetic-energy diagnostics are introduced to quantitatively characterize collision outcomes and to identify phase- and separation-dependent windows of elasticity. To interpret the observed dynamics, we extract effective phase-dependent interaction potentials from collision trajectories, providing a mechanical picture of attraction and repulsion between flat-top solitons. The stability of merged states formed after strongly inelastic collisions is explained by their lower energetic cost, arising from interfacial energetics, where a balance between internal pressure and edge tension plays a central role. A variational analysis based on direct energy minimization supports this picture by revealing robust energetic minima associated with stationary two-dimensional flat-top solitons.

Figures (11)

• Figure 1: Initial and final density surfaces used in the numerical setup. (a) Initial two-Gaussian seed of width $w = 10$, centered at $(x_1,y_1)=(-10,0)$ and $(x_2,y_2)=(22,0)$. (b) Stationary two-FTS state obtained after ITP.
• Figure 2: Central slice comparison of the initial two-Gaussian seed and the stationary two-FTS state produced by ITP, for the same parameters as in Fig. \ref{['fig:twoFTS_initial_final']}.
• Figure 3: Comparison of two collision scenarios obtained with identical parameters, differing only in the initial $x$-position of the second soliton. (a) Elastic scattering for $(x_{2},y_{2})=(30,0)$. (b) Inelastic scattering for $(x_{2},y_{2})=(15,0)$. The qualitative difference arises from the relative phase at the collision point.
• Figure 4: Density and phase evolution along $y=0$ for the elastic collision. (a) Density shows negligible overlap. (b) Phase shows a relative phase close to $\pi$ at the collision point.
• Figure 5: Density and phase evolution along $y=0$ for the inelastic collision. (a) Density shows strong overlap. (b) Phase shows a relative phase close to $0$ at the collision point, leading to deformation and radiation.