Ground States in Spatially Discrete Nonlinear Schr\"odinger Lattices

Abstract

In his seminal work, Weinstein considered the question of the ground states for discrete Schr\"odinger equations with power law nonlinearities, posed on ${\mathbb Z}^d$. More specifically, he constructed the so-called normalized waves, by minimizing the Hamiltonian functional, for fixed power $P$ (i.e. $l^2$ mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit and build upon Weinstein's work in several directions. First, for the normalized waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete NLS evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimization problem. Importantly, this construction works for all values of the parameters, e.g. $l^2$ supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis-Shatah-Strauss/Vakhitov-Kolokolov quantity $\partial_\omega \|\varphi_\omega\|_{l^2}^2$. In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-\'a-vis with the linearized problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameter $\omega$ varies, in contrast with the corresponding continuous NLS model.