Bifurcations of travelling wave solutions in the discrete NLS equations

Abstract

We study discrete nonlinear Schrödinger (NLS) equations, which include the cubic NLS lattice with on-site interactions and the integrable Ablowitz–Ladik lattice. Standing wave solutions are known to exist in the discrete NLS equations outside of the finite spectral band. We study travelling wave solutions which have nonlinear resonances with unbounded linear spectrum. By using center manifold and normal form reductions, we show that a continuous NLS equation with the third-order derivative term is a canonical normal form for the discrete NLS equation near the zero-dispersion limit. Bifurcations of travelling wave solutions near the zero-dispersion limit are analyzed in the framework of the third-order derivative NLS equation. We show that there exists a continuous two-parameter family of single-humped travelling wave solutions in the third-order derivative NLS equation, when it is derived from the integrable Ablowitz–Ladik lattice. On the contrary, there are no single-humped solutions in the third-order derivative NLS equation, when it is derived from the cubic NLS equation with on-site interactions. Nevertheless, we show that there exists an infinite discrete set of one-parameter families of double-humped travelling wave solutions in the latter case. Our results are valid in the neighborhood of the zero-dispersion point on the two-parameter plane of travelling wave solutions.