Stability of KAM tori for nonlinear Schrödinger equation

• Main Authors: Cong, Hongzi, Liu, Jianjun, Yuan, Xiaoping
• Format: eBook Book
• Language: English
• Published: Providence, Rhode Island American Mathematical Society 01.01.2016
• Edition: 1
• Series: Memoirs of the American Mathematical Society
• Subjects: Gross-Pitaevskii equations; Nonlinear wave equations; Perturbation (Mathematics); KAM tori; KAM technique; Stability; <inline-formula content-type="math/mathml"> p p </inline-formula>-tame property; Normal form
• ISBN: 9781470416577; 1470416573
• ISSN: 0065-9266; 1947-6221
• DOI: 10.1090/memo/1134

The authors prove the long time stability of KAM tori (thus quasi-periodic solutions) for nonlinear Schrödinger equation \sqrt{-1}\, u_{t}=u_{xx}-M_{\xi}u+\varepsilon|u|^2u, subject to Dirichlet boundary conditions u(t,0)=u(t,\pi)=0, where M_{\xi} is a real Fourier multiplier. More precisely, they show that, for a typical Fourier multiplier M_{\xi}, any solution with the initial datum in the \delta-neighborhood of a KAM torus still stays in the 2\delta-neighborhood of the KAM torus for a polynomial long time such as |t|\leq \delta^{-\mathcal{M}} for any given \mathcal M with 0\leq \mathcal{M}\leq C(\varepsilon), where C(\varepsilon) is a constant depending on \varepsilon and C(\varepsilon)\rightarrow\infty as \varepsilon\rightarrow0.