Ground states in spatially discrete non-linear Schrödinger models

Abstract

In the seminal work (Weinstein 1999 Nonlinearity 12 673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on Z d . More specifically, he constructed the so-called normalised waves, by minimising 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 these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedke et al 1994 Phys. Rev. Lett. 73 1055 and Laedke et al 1996 Phys. Rev. E 54 4299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation 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 (GSS/VK) quantity ∂ ω ∥ φ ω ∥ 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-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameter ω varies, in contrast with the corresponding continuous NLS model.