J Cuevas, JC Eilbeck and NI Karachalios (pdf copy 224 Kb)
Abstract:
We derive lower bounds on the power of breather solutions
yn(t)=
e-iWtfn,
W>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.
Dynamics of Partial Differential Equations 5 (1), 69-85 (2008)