Discrete Nonlinear Waves

A natural variational problem on a finite type von Neuman algebra is D - > Det(D+m)=exp(tr log(D+m)), where m is a complex parameter and the functional is restricted to some manifold of operators for which the determinant exists. The first nontrivial contribution of this functional is tr(D^4) which becomes essentially the Wilson functional in lattice gauge theory if the variations are restricted to classes of Laplacians. It is a discrete version of the Maxwell-Yang-Mills functional. Critical points lead to nonlinear wave equations. "Journal of Physics A, 29, L595-L600, 1996" Using a variational trick of Aubry one can show that the corresponding coupled map lattice has infinite-dimensional horseshoes "A. Jacobson and O. Knill, Journal of Physics A, 29, L595-L600, 1996". Already primitive experiments in one dimension show interesting dynamics. In the simplest possible situation, where $D$ is invariant under space translations, the time evolution is a Henon type symplectic map.


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