Invariant Cantor Manifolds of Quasi-Periodic Oscillations for a Nonlinear Schrodinger Equation

Abstract

Together they move on a rotational torus of finite or infinite dimension, depending on how many modes are excited. In particular, for every choice J = { j 1 < j 2 < · · · < j n } ⊂ N of n ≥ 1 basic modes there is an invariant linear space E J of complex dimension n which is completely foliated into rotational tori:In addition, each such torus is linearly stable, and all solutions have vanishing Lyapunov exponents. -This is the linear situation.Upon restoring the nonlinearity f the invariant manifolds E J will not persist in their entirety due to resonances among the modes and the strong perturbing effect of f for large amplitudes. We show, however, that in a sufficiently small neighbourhood of the origin a large Cantor subfamily of rotational n -tori persists and is only slightly deformed.

Quasi-periodic solutions for a nonlinear wave equation

“…Our approach and its results are parallel to an investigation of the nonlinear Schrödinger equation iu t = u x x − mu − f (|u| 2 )u on the same interval undertaken by Kuksin and the author in [5]. Hence some parts of the respective expositions are quite similar.…”

“…Hence some parts of the respective expositions are quite similar. But we decided to repeat them anyway so that the reader need not refer to [5] for the essentials.…”

“…Then KAM theory is applicable. This approach was suggested by the author in [6], and carried out for the nonlinear Schrödinger equation by Kuksin and the author in [5]. There some aspects, such as the transformation into normal form, are even simpler than here.…”

“…In section 2 the hamiltonian is written in infinitely many coordinates, which is then put into its partial normal forms in section 3. In section 4 we recall the Cantor Manifold Theorem from [5], which allows to complete the proof of the Main Theorem in section 5.…”

“…by an elementary calculation -see [5]. From now on we focus our attention on the nonlinearity u 3 , since terms of order five or more will not make any difference.…”