Bifurcations of Standing Localized Waves on Periodic Graphs

The nonlinear Schrödinger (NLS) equation is considered on a periodic graph subject to the Kirchhoff boundary conditions. Bifurcations of standing localized waves for frequencies lying below the bottom of the linear spectrum of the associated stationary Schrödinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove the existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graph. We also prove properties of the two families, in particular, positivity and exponential decay. The asymptotic reduction of the two-dimensional discrete map to the stationary NLS equation on an infinite line is discussed in the context of the homogenization of the NLS equation on the periodic graph.