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The Influence of Coupled Wave Networks on the Dynamics of Nonlinear Lattices

Nonlinear lattice models, which describe the dynamics of spatially discretized systems, are an important class of models for understanding the behavior of solid materials, network dynamics, and nonlinear systems. The significance of the spatial topology of these lattice models is well understood, as it can affrect such properties as the steady state dynamics and the spectrum of linear waves. The properties of nonlinear waves in these lattice models and their connection to the network topology, however, is less understood. In this talk we shall present an alternative approach to understand nonlinear wave dynamics and mode mixing in nonlinear lattices. In particular, we shall demonstrate how the nonlinearity of the lattice can be used to define a new network structure which describes the interaction of waves within the lattice. These coupled wave networks are connected to the original spatial network but more directly illustrate the wave dynamics. We shall demonstrate how controlling the structure of a coupled wave network can lead to the control of mode mixing and dynamics, the observation of novel forms of dynamical behavior, and constraints on the system dynamics in real and reciprocal space.