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Nonlinear waves in networks: model reduction for sine-Gordon

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

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