A lower bound for the power of periodic solutions of the defocusing Discrete Nonlinear Schrödinger equation

We derive lower bounds on the power of breather solutions ψn(t) = e −iΩtφn, Ω > 0 of a Discrete Nonlinear Schrödinger Equation with cubic or higher order nonlinearity and site-dependent anharmonic parameter, supplemented with Dirichlet boundary conditions. For the case of a defocusing DNLS, one of the lower bounds depends not only on the dimension of the lattice, the lattice spacing, and the frequency of the periodic solution, but also on the excitation threshold of time periodic and spatially localized solutions of the focusing DNLS, proved by M. Weinstein in Nonlinearity 12, 673–691, 1999. Our simple proof via a direct variational method, makes use of the interpolation inequality proved by Weinstein, and its optimal constant related to the excitation threshold. We also provide existence results (via the mountain pass theorem) and lower bounds on the power of breather solutions for DNLS lattices with sign-changing anharmonic parameter. Numerical studies considering the classical defocusing DNLS, the case of a single nonlinear impurity, as well as a random DNLS lattice are performed, to test the efficiency of the lower bounds.