Multidimensional Discrete Compactons in Nonlinear Schr\"odinger Lattices with Strong Nonlinearity Management

TL;DR

This paper demonstrates the existence and stability of multidimensional lattice compactons in a nonlinear Schrödinger lattice with strong, fast periodic nonlinearity modulations, revealing new stable configurations and their dynamical behaviors.

Contribution

It introduces an averaged dynamical equation with Bessel function couplings that enables exact compacton solutions in multidimensional lattices with time-modulated nonlinearity.

Findings

Single site and vortex compactons are always stable.

Certain multi-site solutions exhibit stability and instability regions.

Time evolution matches averaged dynamics, suggesting observability in BEC experiments.

Abstract

The existence of multidimensional lattice compactons in the discrete nonlinear Schr\"odinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, a new effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kind. These terms allow one to solve, at this averaged level, for exact discrete compacton solution configurations in the corresponding stationary equation. We focus on seven types of compacton solutions: single site and vortex solutions are found to be always stable in the parametric regimes we examined. Other solutions such as double site in- and out-of-phase, four site symmetric and anti-symmetric, and a five site compacton solution are found to have regions of stability and instability in two-dimensional parametric planes, involving variations of the strength of the coupling and of the nonlinearity. We also explore the time evolution of the solutions and compare the dynamics according to the averaged with those of the original dynamical equations without the averaging. Possible observation of compactons in the BEC loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is discussed.