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)

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.

discrete nonlinear Schrodinger equation, traveling wave solutions, Melnikov integrals for differential advance-delay equations, stability of nonlinear waves