An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation

By Jiansheng Geng, Xindong Xu, Jiangong You. 2011

160 citations recorded by the source corpus.

Subfields: Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/78a365ea27955a7c3b4067497a411967392a3f8c

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In Numerical Analysis

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Citations in the corpus

KAM for autonomous quasi-linear perturbations of mKdV (2014, unknown): “For PDEs in higher space dimension the theory has been more recently extended by Bourgain [10], Eliasson–Kuksin [13], and Berti–Bolle [6], Geng–Xu–You [14], Procesi–Procesi [30,29], Wang [32].”
KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation (2014, unknown): “We underline that the goal of the Töplitz-Lipschitz [19], [21], [23] and quasi-Töplitz property [39], [10], [11], [38] is precisely to provide an asymptotic expansion of the perturbed eigenvalues sharp enough to verify the the second order Melnikov conditions.”
A Normal Form for the Schrödinger Equation with Analytic Non-linearities (2010, methodology): “An interesting issue is to see if one can use arithmetic constraints such as those of [11], to simplify those matrices in M which are not self–adjoint.”
Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb {T}^d$ with a multiplicative potential (2013, methodology): “For the cubic NLS in d = 2 the existence of quasi-periodic solutions has been recently proved by Geng-Xu-You [25] via a Birkhoff normal form and a modification of the KAM approach in [21], see also Procesi-Procesi [36], valid in any dimension.”
An Abstract Nash–Moser Theorem and Quasi-Periodic Solutions for NLW and NLS on Compact Lie Groups and Homogeneous Manifolds (2013, background): “Then Geng-Xu-You [16] proved KAM results for the cubic NLS in dimension 2 and Procesi-Procesi [24] in any dimension n and polynomial nonlinearity.”
KAM for Reversible Derivative Wave Equations (2012, methodology): “This is an important novelty with respect to the KAM theorem in [4] which used the conservation of momentum, as [17], [18].”
Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential (2010, methodology): “For the cubic NLS in d = 2 the existence of quasi-periodic solutions has been recently proved by Geng-Xu-You [25] via a Birkhoff normal form and a modification of the KAM approach in [21], see also Procesi-Procesi [36], valid in any dimension.”