Nonlinear waves in networks: model reduction for sine-Gordon

Jean-Guy Caputo (1) , Denys Dutykh (2, 3, 4)

1 LMI - Laboratoire de Mathématiques de l'INSA de Rouen Normandie
2 LAMA - Laboratoire de Mathématiques
3 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc
4 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques

To study how nonlinear waves propagate across Y- and T-type junctions, we consider the 2D sine-Gordon equation as a model and examine the crossing of kinks and breathers. Comparing energies for different geometries reveals that, for small widths, the angle of the fork plays no role. Motivated by this, we introduce a 1D effective model whose solutions agree well with the 2D simulations for kink and breather solutions. These exhibit two different behaviors: a kink crosses if it has sufficient energy; conversely a breather crosses when $v > 1 - omega$, where $v$ and $\omega$ are respectively its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.

Mots clés: kink; breather; sine-Gordon equation; Josephson junction

Domaines: Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Equations aux dérivées partielles [math.AP]; Supraconductivité [cond-mat.supr-con]; Physique mathématique [math-ph]; Physique mathématique [math-ph]; Physique Numérique [physics.comp-ph]; Dynamique des Fluides [physics.flu-dyn]; Formation de Structures et Solitons [nlin.PS]; Systèmes Solubles et Intégrables [nlin.SI]; Analyse numérique [math.NA]

Jean-Guy Caputo, Denys Dutykh. Nonlinear waves in networks: model reduction for sine-Gordon. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2014, 90, pp.022912. ⟨10.1103/PhysRevE.90.022912⟩. ⟨hal-00951705v3⟩