Statistical mechanics of nonlinear wave equations. 3. Metric transitivity for hyperbolic sine-gordon

Abstract

McKean and Vaninsky proved that the canonical measuree ^−H d ^∞ Q d ^∞ P based upon the Hamiltonian$H = \smallint [\tfrac{1}{2}P^2 + \tfrac{1}{2}(Q')^2 + F(Q)]dx$ of the wave equation ∂² Q/∂t ² - ∂² Q/∂x ² +f(Q) = 0 with restoring forcef(Q)=F'(Q) is preserved by the associated flow ofQ andP =Q ^⋅, and they conjectured that metric transitivity prevails,always on the whole line, and likewise on the circleunless f(Q)=Q orf(Q)=shQ. Here, the metric transitivity is proved for the whole line in the second case. The proof employs the beautiful “d'Alembert formula” of Krichever.