Stability of KAM Tori for Nonlinear Schrodinger Equation

ISBN-13: 9781470416577 / Angielski

Hongzi Cong;Jianjun Liu;Xiaoping Yuan

Opis

The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrodinger equation $$sqrt{-1}, u_=u_-M_u+varepsilon|u|^2u,$$ subject to Dirichlet boundary conditions $u(t,0)=u(t,pi)=0$, where $M_$ is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier $M_$, any solution with the initial datum in the $delta$-neighborhood of a KAM torus still stays in the $2delta$-neighborhood of the KAM torus for a polynomial long time such as $|t|leq delta^{-mathcal}$ for any given $mathcal M$ with $0leq mathcalleq C(varepsilon)$, where $C(varepsilon)$ is a constant depending on $varepsilon$ and $C(varepsilon) ightarrowinfty$ as $varepsilon ightarrow0$.

Spis treści

• Introduction and main results
• Some notations and the abstract results
• Properties of the Hamiltonian with $p$-tame property
• Proof of Theorem 2.9 and Theorem 2.10
• Proof of Theorem 2.11
• Proof of Theorem 1.1
• Appendix: technical lemmas
• Bibliography
• Index