IIMAS - FENOMEC
UNAM
Nonlinear Network Wave Equation: Periodic Solutions and Graph Characterizations
Resumen:
We study the discrete nonlinear wave equation in arbitrary finite networks. This is a general model, where the usual continuum Laplacian is replaced by the graph Laplacian. We consider such a wave equation with a cubic on-site nonlinearity which is the discrete Φ4 model, describing a mechanical network of coupled nonlinear oscillators or an electrical network where the components are diodes or Josephson junctions. In the first part, we investigate the extension of the linear normal modes of the graph Laplacian into nonlinear periodic orbits. Normal modes -whose Laplacian eigenvectors are composed uniquely of {1}, {-1,1} or {-1,0,1}- give rise to nonlinear periodic orbits for the discrete Φ4 model. We perform a systematic linear stability (Floquet) analysis of these orbits and show the modes coupling when the orbit is unstable. Then, we characterize graphs having Laplacian eigenvectors in {-1,1} and {-1,0,1} using graph spectral theory. In the second part, we investigate periodic solutions that are exponentially (spatially) localized. Assuming a large amplitude localized initial condition on one node of the graph, we approximate its evolution by the Duffing equation. The rest of the network satisfies a linear system forced by the excited node. This approximation is validated by reducing the discrete Φ4 equation to the discrete nonlinear Schrödinger equation and by Fourier analysis. These results relate nonlinear dynamics to graph spectral theory.
Informes: coloquiomym@gmail.com, o al 5622-3564.