The existence of nonzero periodic travelling wave solutions for a general discrete nonlinear Schrödinger equation (DNLS) on one-dimensional lattices is proved. The DNLS features a general nonlinear term and variable range of interactions going beyond the usual nearest-neighbour interaction. The problem of the existence of travelling wave solutions is converted into a fixed point problem for an operator on some appropriate function space which is solved by means of Schauder’s Fixed Point Theorem.
"Periodic Travelling Wave Solutions of Discrete Nonlinear Schrödinger Equations." J. Appl. Math. 2017 1 - 5, 2017. https://doi.org/10.1155/2017/3694103