Infinite-dimensional non-linear ordinary differential equations and the KAM theory

Abstract

Contents Introduction General notation and definitions Chapter I. Theory of equations with the Prenkel-Kontorova potential §1. Non-unique solubility and insolubility of the Cauchy problem for the standard phase space §2. Introduction of phase spaces and unique solubility in classes of infinitely differentiable and analytic functions §3. Construction of families of periodic solutions and travelling wave solutions, and the Frenkel-Kontorova problem §4. Instability of stationary and periodic solutions in the case of equal masses §5. Stability of stationary solutions and an infinite-dimensional analogue of the KAM theory §6. Proof of Theorem 1.10 §7. Hamiltonian form of an equation with Frenkel-Kontorova potential §8. The Lagrangian form of an equation with Frenkel-Kontorova potential Chapter II. Theory of an equation with Fermi-Pasta-Ulam non-linearity §1. Construction of families of periodic solutions and of solutions generating oscillations of a finite discrete string §2. Introduction of phase spaces and unique solubility of an equation in a neighbourhood of the zero solution §3. Stability of the zero solution and an infinite-dimensional analogue of the KAM theory §4. Proof of Theorem 2.4 §5. On stability with respect to a non-autonomous dynamical system with discrete time §6. Local non-unique solubility and insolubility of an equation in the standard phase space Chapter III. Infinite-dimensional ordinary differential equations of general form §1. Formal unique solubility and uniqueness of analytic solutions §2. Stability of stationary solutionsBibliography