Discrete sine-Gordon dynamics on networks

Denys Dutykh (1) , Jean-Guy Caputo (2)

1 LAMA - Laboratoire de Mathématiques
2 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie

In this study we consider the sine-Gordon equation formulated on domains which are not locally homeomorphic to any subset of the Euclidean space. More precisely, we formulate the discrete dynamics on trees and graphs. Each edge is assumed to be a 1D uniform lattice with end points identified with graph vertices. A special treatment is needed at the junctions in order to couple 1D lattices into a global communicating network. Our approach is based on considering the local conservation properties. Some preliminary numerical results are shown on a simple graph containing four loops. These results show the performance of the scheme in non-trivial realistic conditions.

Mots clés: kink solution; networks; sine-Gordon

Domaines: Physique mathématique [math-ph]; Physique Numérique [physics.comp-ph]; Systèmes Solubles et Intégrables [nlin.SI]; Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Analyse numérique [cs.NA]; Modélisation et simulation; Formation de Structures et Solitons [nlin.PS]

Denys Dutykh, Jean-Guy Caputo. Discrete sine-Gordon dynamics on networks. 2015. ⟨hal-01160840v1⟩