Metastability of the Nonlinear Wave Equation: Insights from Transition State Theory

This paper is concerned with the longtime dynamics of the nonlinear wave equation in one-space dimension,(Formula presented.)where κ> 0 is a parameter and V(u) is a potential bounded from below and growing at least like u² as | u| → ∞. Infinite energy solutions of this equation preserve a natural Gibbsian invariant measure, and when the potential is double-welled, for example when V(u)=14(1-u2)2, there is a regime such that two small disjoint sets in the system’s phase-space concentrate most of the mass of this measure. This suggests that the solutions to the nonlinear wave equation can be metastable over these sets, in the sense that they spend long periods of time in these sets and only rarely transition between them. Here, we quantify this phenomenon by calculating exactly via transition state theory (TST) the mean frequency at which the solutions of the nonlinear wave equation with initial conditions drawn from its invariant measure cross a dividing surface lying in between the metastable sets. We also investigate numerically how the mean TST frequency compares to the rate at which a typical solution crosses this dividing surface. These numerical results suggest that the dynamics of the nonlinear wave equation is ergodic and rapidly mixing with respect to the Gibbs invariant measure when the parameter κ in small enough. In this case, successive transitions between the two regions are roughly uncorrelated and their dynamics can be coarse-grained to jumps in a two-state Markov chain whose rate can be deduced from the mean TST frequency. This is a regime in which the dynamics of the nonlinear wave equation displays a metastable behavior that is not fundamentally different from that observed in its stochastic counterpart in which random noise and damping terms are added to the equation. For larger κ, however, the dynamics either stops being ergodic, or its mixing time becomes larger than the inverse of the TST frequency, indicating that successive transitions between the metastable sets are correlated and the coarse-graining to a Markov chain fails.