Modulational and Parametric Instabilities of the Discrete Nonlinear Schrödinger Equation

Abstract

We examine the modulational and parametric instabilities arising in a non-autonomous, discrete nonlinear Schr{ö}dinger equation setting. The principal motivation for our study stems from the dynamics of Bose-Einstein condensates trapped in a deep optical lattice. We find that under periodic variations of the heights of the interwell barriers (or equivalently of the scattering length), additionally to the modulational instability, a window of parametric instability becomes available to the system. We explore this instability through multiple-scale analysis and identify it numerically. Its principal dynamical characteristic is that, typically, it develops over much larger times than the modulational instability, a feature that is qualitatively justified by comparison of the corresponding instability growth rates.

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• Heteroclinic structure of parametric resonance in the nonlinear Schrödinger equation
• Modulational instability of two-component Bose-Einstein condensates in an optical lattice
• Dynamical instability of a Bose-Einstein condensate with higher-order interactions in an optical potential through a variational approach
• Dynamic and Static Excitations of a Classical Discrete Anisotropic Heisenberg Ferromagnetic Spin Chain
• Faraday waves in BEC with engineering three-body interactions
• Instability of Holographic Superfuids in Optical Lattice
• Application of the Feshbach-resonance management to a tightly confined Bose-Einstein condensate
• Optical Lattices: Theory
• Unstaggered-staggered solitons on one- and two-dimensional two-component discrete nonlinear Schrödinger lattices