Nonlinear stability in integrable Hamiltonian systems.

摘要

The stability of periodic solutions of partial differential equations has been an area of increasing interest in the last decade. In this thesis, a new method for investigating the (nonlinear) orbital stability of periodic solutions of integrable Hamiltonian systems is presented. The method is demonstrated on the KdV equation, proving that all periodic finite-genus solutions are orbitally stable with respect to subharmonic perturbations (perturbations that have period equal to an integer multiple of the period of the amplitude of the solution). Also, a reduced form of the method is applied to the NLS and mKdV equations, establishing the orbital stability of elliptic solutions of the defocusing NLS equation and traveling wave solutions of the defocusing mKdV equation, both with respect to subharmonic perturbations.