Stability and Interaction Dynamics of Solitons in Spatially Engineered High-Order Nonlinear Media
We study spatial solitons and their interaction dynamics in nonlinear optical media with spatially engineered refractive index and competing cubic-quintic (CQ) nonlinear profiles, using the variational approximation (VA), the hybrid variational approximation (HVA), and direct numerical simulations. The model was implemented in a symmetric step-index planar dielectric waveguide with a core exhibiting competing CQ nonlinearity and cladding layers possessing only a cubic nonlinear response. For stationary states, the Gaussian VA predicts two types of $N(μ)$ characteristic curves, where $N$ is soliton norm and $μ$ is propagation constant, separated by a boundary surface in parameter space, and numerical calculations reveal the same two types. Below this surface, the VA agrees with the numerical results mainly at low powers, while pronounced deviations in the profile and stability appear at high powers. Above the surface, the variational and numerical curves retain the same qualitative form, and a super-Gaussian ansatz accurately describes the high-power flat-top solitons. Soliton collisions produce four post-interaction regimes: Oscillation, Molecular, Splitting, and Breakup. The HVA reproduces the first three over a broad power range, including collisions involving flat-top solitons. Its main limitation arises in the Breakup regime, where strong radiation leaves the guiding region and cannot be represented by the adopted HVA ansatz. Nevertheless, for solitons associated with the second type of characteristic curves, the HVA still captures the breakup dynamics qualitatively. Thus, the HVA provides an efficient description of complex soliton interactions at a substantially lower computational cost than direct numerical simulations.