Analytic Lyapunov exponents in a classical nonlinear field equation

It is shown that the nonlinear wave equation partial derivative(t)(2)phi-pa rtial derivative(x)(2)phi-mu(0)partial derivative(x)(partial derivative(x)p hi)(3)=0, which is the continuum limit of the Fermi-Pasta-Ulam beta model, has a positive Lyapunov exponent lambda(1), whose analytic energy dependenc e 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 w ithin the framework of a Riemannian description of Hamiltonian chaos. We al so discuss a difficulty of the statistical mechanical treatment of this cla ssical field system, which is absent in the dynamical description.