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Abstract

The purpose of this paper is to establish the existence and spectral stability, with respect to perturbations of the same period, of double-periodic standing waves for the nonlinear focusing Schr\"odinger equation posed on the bi-dimensional torus. We first show that such double-periodic solutions can be constructed via local and global bifurcation theory, under the assumption that the kernel of the linearized operator around the equilibrium solution is one-dimensional. In addition, we prove that these local and global solutions minimize an appropriate variational problem, which enables us to derive spectral properties of the linearized operator about the periodic wave. Finally, we establish the spectral stability of small-amplitude periodic waves by applying the techniques developed in \cite{KapitulaKevrekidisSandstedeII} and \cite{KapitulaKevrekidisSandstedeI}.