Mechanics not on a manifold

Abstract

The free scalar field is studied on the Y-junction of three semi-infinite axes which is the simplest example of a non-manifold space. It is shown that the transition rules for this system uniquely follow from conservation of energy and charge. A discrete version of the model gives the same result.

Nonlinear waves in networks: Model reduction for the sine-Gordon equation

“…On the other hand, for frequencies close to one, the breather crosses even for small velocities. This situation is close to the linear case for which we expect always some energy transfer to the other branch [3].…”

“…For these linear systems, the scattering formalism can be employed and this gives the reflection and transmission coefficients for a harmonic wave. This is detailed specifically for a Yjunction and for the Klein-Gordon linear wave equation in [3].…”

Wave dynamics on networks: method and application to the sine-Gordon equation

“…Moreover, this condition has to be completed by the "charge" conservation property adopted in previous studies of the Klein-Gordon [5,28] and sG [8] equations:…”

“…To our knowledge, Hamiltonian problems on non-manifolds have not been systematically studied. A notorious exception is the work [5], where wave scattering in the Klein-Gordon(-Fok) equation was investigated on a domain consisting of three semi-infinite straight lines having one common point. We can also mention the publication [6] where the Benjamin-Bona-Mahony (BBM) equation was considered on a tree.…”

“…The transition rules between the adjacent lattices at junction points follow from the discretization of the local conservation laws. This approach was already used in [5,8]. We expect that the limit of the lattice parameter ∆ → 0 will provide us with the continuous version of the Hamiltonian mechanics on non-manifolds.…”