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

Geng, Jiao; Xu, Xindong; You, Jiangong

doi:10.1016/j.aim.2011.01.013

Cited by 122 publications

(88 citation statements)

References 19 publications

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“…For high dimensional NLS see the milestone work by Eliasson and Kuksin [14], where they found and defined a Töplitz-Lipschitz property and used it to control the shift of the normal frequencies. See [16] and [37] for recent development in nd-NLS. For nd-beam equations see [18] and [19] for the nonlinearity g(u) and see [11] and [12] for more general nonlinearity g(x, u).…”

Section: Related Resultsmentioning

confidence: 99%

Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

Wang

Liang

2017

Nonlinearity

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Abstract. In this paper we prove an infinite dimensional KAM theorem, in which the assumptions on the derivatives of perturbation in [22] are weakened from polynomial decay to logarithmic decay. As a consequence, we apply it to 1d quantum harmonic oscillators and prove the reducibility of a linear harmonic oscillator,perturbed by a quasiperiodic in time potential V (x, ωt; ω) with logarithmic decay. This entails the pure-point nature of the spectrum of the Floquet operator K, whereand the potential V (x, θ; ω) has logarithmic decay as well as its gradient in ω.

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Section: Related Resultsmentioning

confidence: 99%

Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

Wang

Liang

2017

Nonlinearity

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“…Geng-You [12,15,16] and Eliasson-Kuksin [11] constructed quasi-periodic solutions of higher dimensional Hamiltonian PDEs respectively by method from an infinite dimensional KAM theory. For the completely resonant cubic Schrödinger equation on a torus T d , the existence of quasi-periodic solutions were proved by Procesi and Procesi [23].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces

Zhang¹,

Rui²

2017

J. Nonlinear Sci. Appl.

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In this paper, d-dimensional (dD) quasi-periodically forced nonlinear Schrödinger equation with a general nonlinearityunder periodic boundary conditions is studied, where M ξ is a real Fourier multiplier and ε is a small positive parameter, φ(t) is a real analytic quasi-periodic function in t with frequency vector ω = (ω 1 , ω 2 . . . , ω m ), and h(|u| 2 ) is a real analytic function near u = 0 with h(0) = 0. It is shown that, under suitable hypothesis on φ(t), there are many quasi-periodic solutions for the above equation via KAM theory.

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“…Previously, quasi-periodic solutions were constructed using partial Birkhoff normal forms for the resonant Schrödinger equation in the presence of the cubic nonlinearity in dimensions 1 and 2 [7,20,21]. These algebraic normal form constructions use in an essential way the specifics of the resonance geometry generated by the cubic nonlinearity, see the Appendix in sect.…”

Section: Spectral Methods In Pde 567mentioning

confidence: 99%

Spectral Methods in PDE

Wang

2010

Milan J. Math.

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

Cited by 122 publications

References 19 publications

Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

Reducibility of 1D quantum harmonic oscillator perturbed by a quasiperiodic potential with logarithmic decay

Quasi-periodic solutions of Schrodinger equations with quasi-periodic forcing in higher dimensional spaces

Spectral Methods in PDE

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