Analytic Lyapunov exponents in a classical nonlinear field equation

It is shown that the nonlinear wave equation ${\partial }_t^2\phi -{\partial }_x^2\phi -{\mu }_0{\partial }_x({\partial }_x\phi {)}^3=0,$ 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.