Analytic lyapunov exponents in a classical nonlinear field equation

It is shown that the nonlinear wave equation partial differential(2)(t)straight phi- partial differential2xstraight phi-&mgr;(0) partial differential(x)( partial differential(x)straight phi)(3)=0, which is the continuum limit of the Fermi-Pasta-Ulam 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.