Quasi-Periodic Solutions for 1D Schrödinger Equations with Higher Order Nonlinearity

Liang, Zhenwen; You, Jiangong

doi:10.1137/s0036141003435011

Cited by 55 publications

(42 citation statements)

References 14 publications

(22 reference statements)

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“…For the KAM results with bounded perturbations see [17,25,27,28,31,32] for 1d-NLS. 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.…”

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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“…Denote G ijd = G ijdijd and G i = G iiiiii . From [9], if the index set I := {n 1 < n 2 < · · · < n b } satisfies…”

Section: Partial Birkhoff Normal Formmentioning

confidence: 99%

“…We apply the method of [9] to define admissible index sets. For each index set I, define Then the admissible index set can be defined as follows.…”

Section: Partial Birkhoff Normal Formmentioning

confidence: 99%

“…We note that the 1D nonlinear Schrödinger equation (3) is completely integrable, hence one can perturb any finite number of Fourier modes to obtain small amplitude quasi-periodic solutions, which is however not the case if the nonlinearities are of high order. For Schrödinger equations with high order nonlinearities, Liang and You [9] obtained quasi-periodic solutions corresponding to any finite number of admissible Fourier modes for the Schrödinger equation…”

Section: Introductionmentioning

confidence: 99%

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Quasi-periodic solutions for a Schrödinger equation with a quintic nonlinear term depending on the time and space variables

2018

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This article is devoted to the study of a nonlinear Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term. It is proved that the equation admits small-amplitude, linearly stable, real analytic, and quasi-periodic solutions for most values of frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some sixth order Birkhoff normal form. We show an infinite-dimensional KAM theorem for non-autonomous Schrödinger equations and apply the theorem to prove the existence of quasi-periodic solutions.

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“…So, using Lemma 3.3 in [26], we get, for (ϑ; ω) ∈ Θ(σ ν+1 ) × Ω , 20) where C := CC * ρ −1 . Moreover, by (2.10) ν and (2.11) ν , it follows that…”

Section: Reducibility Of Schrödinger Equationsmentioning

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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Quasi-Periodic Solutions for 1D Schrödinger Equations with Higher Order Nonlinearity

Cited by 55 publications

References 14 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 for a Schrödinger equation with a quintic nonlinear term depending on the time and space variables

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

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