Analytic Lyapunov exponents in a classical nonlinear field equation

It is shown that the nonlinear wave equation $\partial^2_t \phi -\partial^2_x \phi -\mu_0 \partial_x(\partial_x \phi)^3=0$, 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.