  It is well known that a generic small perturbation of a Liouville-integrable
Hamiltonian system causes breakup of resonant and near-resonant invariant tori.
A general approach to the simple resonance case in the convex real-analytic
setting is developed, based on a new technique for solving the Hamilton-Jacobi
equation. It is shown that a generic perturbation creates in the core of a
resonance a partially hyperbolic lower-dimensional invariant torus, whose
Lagrangian stable and unstable manifolds, described as global solutions of the
Hamilton-Jacobi equation, split away from this torus at exponentially small
angles. Optimal upper bounds with best constants are obtained for exponentially
small splitting in the general case.
