S. Le Coz, D.E. Pelinovsky and G. Schneider
Traveling waves in periodic metric graphs via spatial dynamics
Abstract:
The purpose of this work is to introduce a concept of traveling waves in the setting
of periodic metric graphs. It is known that the nonlinear Schrodinger (NLS) equation
on periodic metric graphs can be reduced asymptotically on long but finite time intervals
to the homogeneous NLS equation, which admits traveling solitary wave solutions. In order
to address persistence of such traveling waves beyond finite time intervals, we formulate
the existence problem for traveling waves via spatial dynamics. There exist no spatially
decaying (solitary) waves because of an infinite-dimensional center manifold in the spatial
dynamics formulation. Existence of traveling modulating pulse solutions which are solitary
waves with small oscillatory tails at very long distances from the pulse core is proven
by using a local center-saddle manifold. We show that the variational formulation fails
to capture existence of such modulating pulse solutions even in the singular limit of zero
wave speeds where true (standing) solitary waves exist. Propagation of a traveling solitary
wave and formation of a small oscillatory tail outside the pulse core is shown in numerical
simulations of the NLS equation on the periodic graph.
Keywords:
nonlinear Schrodinger equation, periodic metric graphs, traveling waves, spatial dynamics, invariant
manifolds, oscillating trails.