Discrete structures equivalent to nonlinear Dirichlet and wave equations

The Amann-Conley-Zehnder (ACZ) reduction is a global Lyapunov-Schmidt reduction for PDEs based on spectral decomposition. ACZ has been applied in conjunction to diverse topological methods, to derive existence and multiplicity results for Hamiltonian systems, for elliptic boundary value problems, and for nonlinear wave equations. Recently, the ACZ reduction has been translated numerically for semilinear Dirichlet problems and for modeling molecular dynamics, showing competitive performances with standard techniques. In this paper, we apply ACZ to a class of nonlinear wave equations in T(n), attaining to the definition of a finite lattice of harmonic oscillators weakly nonlinearly coupled exactly equivalent to the continuum model. This result can be thought as a thermodynamic limit arrested at a small but finite scale without residuals. Reduced dimensional models reveal the macroscopic scaled features of the continuum, which could be interpreted as collective variables.