D.E. Pelinovsky, T.R.O. Melvin and A.R. Champneys
One-parameter localized traveling waves in nonlinear Schrodinger lattices
Physica D 236, 22-43 (2007)
Abstract:
We address traveling single-humped localized solutions in the
spatial discretizations of the nonlinear Schr\"{o}dinger (NLS)
equation. By using the Implicit Function Theorem for solution of
the differential advance-delay equation in exponentially weighted
spaces, we develop a mathematical technique for analysis of
persistence of traveling solutions. The technique is based on a
number of assumptions on the linearization spectrum, which are
checked numerically in the general case. We apply the technique to
a wide class of discrete NLS equations with general cubic
nonlinearity which includes the Salerno model, the translationally
invariant and the Hamiltonian NLS lattices as special cases. We
show that the traveling solutions terminate in the Salerno model
and they persist generally in the other two NLS lattices as a
one-parameter family of the relevant differential advance-delay
equation. These results are found to be in a close correspondence
with numerical approximations of traveling solutions with zero
radiation tails. Analysis of persistence also predicts the
spectral stability of the one-parameter family of traveling
solutions in the time evolution of the discrete NLS equation.
Keywords:
discrete nonlinear Schrodinger equation, traveling wave solutions, Melnikov integrals for
differential advance-delay equations, stability of nonlinear waves