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

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.

💡 Research Summary

This paper presents a comprehensive numerical study of collisions between two-dimensional flat‑top solitons (FTSs) governed by the cubic‑quintic nonlinear Schrödinger equation (CQ‑NLSE). The authors focus on how the relative phase of the solitons at the moment of impact controls whether the interaction is essentially elastic, weakly inelastic, or strongly inelastic leading to a merged, long‑lived state.

The model equation is
(i\psi_t + g_1(\psi_{xx}+\psi_{yy}) + g_2|\psi|^2\psi + g_3|\psi|^4\psi = 0),
with (g_2>0) (self‑focusing cubic term) and (g_3<0) (self‑defocusing quintic term). The competition between these nonlinearities produces solitons whose interior density is nearly constant while the edges are sharp, giving them liquid‑like properties such as an effective surface tension.

Stationary FTSs are obtained by imaginary‑time propagation (ITP) using a split‑step Fourier scheme. The authors fix the norm (N=80) and choose parameters (g_1=1/2), (g_2=4), (g_3=-4). Starting from two broad Gaussian seeds, the ITP converges to a radially symmetric flat‑top profile. By superposing two such stationary solutions at a controlled separation (\Delta x) and imprinting a uniform phase on one of them, they generate initial two‑soliton configurations for real‑time evolution.

Collision simulations are performed on a (1024\times1024) grid with periodic boundaries. One soliton is initially at rest while the other receives a small momentum kick, ensuring a head‑on encounter along the x‑axis. The only variable altered between runs is either the initial separation (which determines the accumulated phase difference at impact) or an explicit phase offset (\Delta\phi) applied to one soliton.

The key observation is that the relative phase at the collision point dictates the outcome. When the phase difference is close to (\pi) (out‑of‑phase), destructive interference suppresses overlap, the solitons pass through each other with negligible deformation, and the kinetic energy before and after the collision remains essentially unchanged. Conversely, when the phase difference is near zero (in‑phase), the solitons strongly overlap, emit radiation, and often coalesce into a single, broader flat‑top structure.

To quantify elasticity, the authors introduce a kinetic‑energy diagnostic
(\Delta KE = K_E^{\text{after}} - K_E^{\text{before}}).
Values of (\Delta KE) close to zero identify nearly elastic collisions, while larger positive values signal inelastic energy transfer to radiation and internal modes. Systematic scans of (\Delta KE) versus initial separation reveal an intermediate “low‑(\Delta KE)” window corresponding to elastic scattering, flanked by regions of higher (\Delta KE) where collisions become increasingly inelastic. A similar periodic dependence is observed when (\Delta KE) is plotted against the imposed phase offset (\Delta\phi), confirming that the phase accumulated during propagation is the primary control parameter.

Beyond phenomenology, the authors extract an effective phase‑dependent interaction potential (V_{\text{eff}}(\Delta x,\Delta\phi)) directly from the soliton trajectories. The potential is repulsive for out‑of‑phase configurations and attractive for in‑phase ones, with a distance dependence shaped by the soliton’s edge tension. This mechanical picture extends earlier one‑dimensional analyses to two dimensions, where curvature of the interface and additional transverse degrees of freedom play a role.

Strongly inelastic collisions often produce a merged state that persists for long simulation times. To explain its stability, the paper combines interfacial energetics with a Young‑Laplace‑type balance: the internal pressure generated by the bulk nonlinearity is balanced by the surface tension of the soliton’s edge, yielding a relation (P = 2\sigma/R) (with (R) an effective radius). A variational calculation based on direct energy minimization at fixed norm shows that the merged configuration corresponds to a true energetic minimum, not merely a transient excitation. Consequently, the merged flat‑top soliton is a stable stationary solution of the CQ‑NLSE.

In summary, the study demonstrates that:

1. The relative phase at impact is the dominant factor separating elastic from inelastic collisions of 2D flat‑top solitons.
2. Kinetic‑energy diagnostics provide a clear quantitative measure of elasticity and reveal alternating windows of elastic and inelastic behavior as a function of initial separation or phase offset.
3. An effective interaction potential extracted from the dynamics offers a simple mechanical interpretation of phase‑controlled attraction and repulsion.
4. Merged states formed after strong inelastic collisions are energetically favored due to a balance between bulk pressure and edge tension, and they correspond to stable minima of the system’s Hamiltonian.

These findings enrich the understanding of soliton interactions in non‑integrable, higher‑dimensional settings and furnish practical tools for controlling soliton collisions in experimental platforms such as nonlinear optics, Bose‑Einstein condensates, and plasma physics.