KAM for the nonlinear Schrödinger equation

By L. Eliasson, Sergei B. Kuksin. 2010

327 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/b1f97d8efb12ed21372637099e000b0e8fcf70fb

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

An infinite dimensional KAM theorem and its application to the two dimensional cubic Schrödinger equation (2011, methodology): “Some θ -dependent quadratic terms ∑ |n| =|m| Pnm(θ)znz̄m will be kept in the normal form part, thus the KAM theorem in [11] cannot be applied directly to our case.”
MODIFIED SCATTERING FOR THE CUBIC SCHRÖDINGER EQUATION ON PRODUCT SPACES AND APPLICATIONS (2013, background): “In this case, many different long-time behaviors can be sustained even on arbitrarily small open sets around zero, ranging from KAM tori [12, 38, 74, 80] to heteroclinic orbits [31, 47] and coherent out-of-equilibrium frequency dynamics(2) [39].”
Growth of Sobolev norms in the cubic defocusing nonlinear Schr\"odinger equation (2012, background): “8, EK10] since, in these papers, the authors study the existence of quasi-periodic solutions for the nonlinear Schrodinger equation in the 2-dimensional torus [Bou98] and in a torus of any dimension [EK10]. Nevertheless, they consider slightly dierent equations containing a convolution potential. 2 ”
Quasi-periodic solutions with Sobolev regularity of NLS on T^d with a multiplicative potential (2010, methodology): “We also remark that, in the last years, the KAM approach has been extended by Eliasson-Kuksin [21] for nonlinear Schrödinger equations on Td with a convolution potential and analytic nonlinearities.”
KAM for the nonlinear beam equation (2016, methodology): “This has become quite common in KAM and was also used in [17].”
Reducibility of first order linear operators on tori via Moser's theorem (2018, background): “Regarding reducible KAM theory for non linear PDEs the literature is quite vast, we mention the classical papers [27],[41],[29],[14] for PDEs on the circle, [21],[17],[22],[37],[16] for PDEs on T n .”
A KAM algorithm for the resonant non-linear Schrödinger equation (2012, methodology): “KAM results for PDEs in dimension d > 1 are few and relatively recent, see for instance [11], and in particular the paper [8] which studies an NLS with external parameters.”