Analytic Lyapunov exponents in a classical nonlinear field equation

Abstract

It is shown that the nonlinear wave equation ${\partial }_t^2\varphi -{\partial }_x^2\varphi -{\mu }_0{\partial }_x({\partial }_x\varphi {)}^3=0$, which is the continuum limit of the Fermi-Pasta-Ulam (FPU) beta model, has a positive Lyapunov exponent lambda_1, whose analytic energy dependence is given. The result (a first example for field equations) is achieved by evaluating the lattice-spacing dependence of lambda_1 for the FPU model within the framework of a Riemannian description of Hamiltonian chaos. We also discuss a difficulty of the statistical mechanical treatment of this classical field system, which is absent in the dynamical description.

Comments: 4 pages, 1 figure