Multidimensional discrete compactons in nonlinear Schrödinger lattices with strong nonlinearity management

Abstract

The existence of multidimensional lattice compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, an effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kinds. We show that 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 antisymmetric, 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 equations with those of the original dynamical system. The possible observation of compactons in Bose-Einstein condensates loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is also discussed

Binary matter-wave compactons induced by inter-species scattering length modulations

“…In particular, the existence of one-dimensional compactons in BEC arrays [5] and in binary BEC mixtures with time modulated intra-spices scattering length [6], as well as, the existence of compactons and vortexcompactons (e.g. vortices on a compact support) in two-dimensional DNLSE [7], were recently demonstrated.…”

“…From this point of view appears particularly interesting the theoretical work done on compactons of the discrete nonlinear Schrödinger equation (DNLSE), a model that is linked to physical systems such as nonlinear optical waveguides arrays and Bose-Einstein condensates (BEC) trapped in deep optical lattices (OL). In these contexts, it has been demonstrated that when subjected to periodic time dependent rapid modulations of the nonlinearity, also called strong nonlinearity management (SNM) limit, the DNLSE can support stable compactons of different types [5,6,7]. In particular, the existence of one-dimensional compactons in BEC arrays [5] and in binary BEC mixtures with time modulated intra-spices scattering length [6], as well as, the existence of compactons and vortexcompactons (e.g.…”

Theoretical studies of Bose-Hubbard and discrete nonlinear Schrödinger models : Localization, vortices, and quantum-classical correspondence

“…4.6.1). More recently it was shown that compactons also can exist in a DNLS model with a fast, periodic modulation of the nonlinearity, both in one- [35] and several-dimensional [36] lattices. By time averaging over the fast modulation, the (nearest-neighbor) coupling becomes density dependent, and it will also vanish for certain amplitudes which allow for compactons.…”

“…Note that this is the experimental scheme discussed in Refs. [35,36], however within a DNLS context, for realizing lattice compactons (see Sec. 1.3.4).…”

“…the material's refractive index n will change as 36) where n 0 is the ordinary low-intensity refractive index, and n 2 the Kerr index. The 'original' electro-optical Kerr effect, which was discovered by the Scottish physicist John Kerr in the 1870's [91], describes the same dependence for the refractive index as in (2.36) but on static electric fields.…”