# Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices

## TL;DR

This paper demonstrates the stabilization of two-dimensional matter-wave solitons and vortices in a cubic-quintic nonlinear lattice, expanding the understanding of localized modes in nonlinear media with controllable periodic nonlinearities.

## Contribution

It introduces a novel nonlinear lattice with independently controllable periods for cubic and quintic nonlinearities, enabling stable 2D solitons and vortices in the cubic-quintic model.

## Key findings

- Stable fundamental solitons follow the Vakhitov-Kolokolov criterion.
- Vortex solitons are stable only under specific period conditions.
- Stable localized modes can be realized in Bose-Einstein condensates and optical media.

## Abstract

The nonlinear lattice---a new and nonlinear class of periodic potentials---was recently introduced to generate various nonlinear localized modes. Several attempts failed to stabilize two-dimensional (2D) solitons against their intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wave solitons and vortices in an extended setting---the cubic and quintic model---by introducing another nonlinear lattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity, therefore making a fully `nonlinear quasi-crystal'. A variational approximation based on Gaussian ansatz is developed for the fundamental solitons and in particular, their stability exactly follows the inverted \textit{Vakhitov-Kolokolov} stability criterion, whereas the vortex solitons are only studied by means of numerical methods. Stability regions for two types of localized mode---the fundamental and vortex solitons---are provided. A noteworthy feature of the localized solutions is that the vortex solitons are stable only when the period of the quintic nonlinear lattice is the same as the cubic one or when the quintic nonlinearity is constant, while the stable fundamental solitons can be created under looser conditions. Our physical setting (cubic-quintic model) is in the framework of the Gross-Pitaevskii equation (GPE) or nonlinear Schr\"{o}dinger equation, the predicted localized modes thus may be implemented in Bose-Einstein condensates and nonlinear optical media with tunable cubic and quintic nonlinearities.