Bifurcations of standing localized waves on periodic graphs

Abstract

The nonlinear Schrodinger (NLS) equation is considered on a periodic metric 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 Schrodinger equation are considered by using analysis of two-dimensional discrete maps near hyperbolic fixed points. We prove existence of two distinct families of small-amplitude standing localized waves, which are symmetric about the two symmetry points of the periodic graphs. 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 metric graph.

23 pages

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• Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph
• Nonlinear dynamics on branched structures and networks
• Symmetry breaking in two-dimensional square grids: persistence and failure of the dimensional crossover
• An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity
• Dynamical and variational properties of the NLS-${\delta }_s^{\prime }$ equation on the star graph
• Asymptotic approximation for the solution to a semi-linear parabolic problem in a thin star-shaped junction
• Spectral properties of nonlinear Schrödinger equation on a ring
• NLS Bifurcations on the bowtie combinatorial graph and the dumbbell metric graph