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D.E. Pelinovsky and G. Schneider

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Bifurcations of standing localized waves on periodic graphs

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Annales Henri Poincare 18, 1185-1211 (2017)

**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.

**Keywords**:

Nonlinear Schrodinger equation; periodic quantum graphs, standing wave solutions,
existence and bifurcations, discrete maps, hyperbolic points, approximations