Ground-state properties of small-size nonlinear dynamical lattices

We investigate the ground state of a system of interacting particles in small nonlinear lattices with M >or=3 sites, using as a prototypical example the discrete nonlinear Schrödinger equation that has been recently used extensively in the contexts of nonlinear optics of waveguide arrays and Bose-Einstein condensates in optical lattices. We find that, in the presence of attractive interactions, the dynamical scenario relevant to the ground-state and the lowest-energy modes of such few-site nonlinear lattices reveals a variety of nontrivial features that are absent in the large/infinite lattice limits: the single-pulse solution and the uniform solution are found to coexist in a finite range of the lattice intersite coupling where, depending on the latter, one of them represents the ground state; in addition, the single-pulse mode does not even exist beyond a critical parametric threshold. Finally, the onset of the ground-state (modulational) instability appears to be intimately connected with a nonstandard ("double transcritical") type of bifurcation that, to the best of our knowledge, has not been reported previously in other physical systems.