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83
Quasiperiodic solutions of nonlinear random Schrödinger equations
 J. EUR. MATH. SOC
, 2008
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Pure point spectrum of the Floquet Hamiltonian for the quantum harmonic oscillator under time quasiperiodic perturbations
, 2006
"... We prove that the 1−d quantum harmonic oscillator is stable under spatially localized, time quasiperiodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by EnssVeselic in their 1983 paper [EV] in the general quasiperiodic setting. The motiv ..."
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Cited by 11 (1 self)
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We prove that the 1−d quantum harmonic oscillator is stable under spatially localized, time quasiperiodic perturbations on a set of Diophantine frequencies of positive measure. This proves a conjecture raised by EnssVeselic in their 1983 paper [EV] in the general quasiperiodic setting. The motivation of the present paper also comes from construction of quasiperiodic solutions for the corresponding nonlinear equation.
Periodic solutions for a class of nonlinear partial differential equations in higher dimension
, 2008
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Resonant dynamics for the quintic nonlinear Schrödinger equation
 Ann. Inst. H. Poincaré Anal. Non Linéaire
, 2012
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A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations ∗
, 2010
"... Abstract: We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vectorfield, which could be applied to a large class of Hamiltonian PDEs containing the derivative ∂x in the perturbation. Especially, in this range of application lie a class of derivative nonli ..."
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Abstract: We establish an abstract infinite dimensional KAM theorem dealing with unbounded perturbation vectorfield, which could be applied to a large class of Hamiltonian PDEs containing the derivative ∂x in the perturbation. Especially, in this range of application lie a class of derivative nonlinear Schrödinger equations with Dirichlet boundary conditions and perturbed BenjaminOno equation with periodic boundary conditions, so KAM tori and thus quasiperiodic solutions are obtained for them.
Renormalization Group and the Melnikov Problem for PDE's
 COMM. MATH. PHYS
, 2001
"... We give a new proof of persistence of quasiperiodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dime ..."
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We give a new proof of persistence of quasiperiodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of smallamplitude, quasiperiodic solutions of nonlinear wave equations with periodic boundary conditions.
Quasiperiodic solutions in a nonlinear Schrödinger equation
 J. Differential Equations
, 2007
"... Abstract. In this paper, onedimensional (1D) nonlinear Schrödinger equation iut − uxx + mu + u  4 u = 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescibed interger N> 1, the equation admits a Whitney smooth family of sma ..."
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Cited by 7 (2 self)
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Abstract. In this paper, onedimensional (1D) nonlinear Schrödinger equation iut − uxx + mu + u  4 u = 0 with the periodic boundary condition is considered. It is proved that for each given constant potential m and each prescibed interger N> 1, the equation admits a Whitney smooth family of smallamplitude, time quasiperiodic solutions with N Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an improved KAM method. Consider a nonlinear Schrödinger equation
Modified scattering for the cubic Schrödinger equation on product spaces and applications
, 2013
"... Abstract. We consider the cubic nonlinear Schrödinger equation posed on the spatial domain R × Td. We prove modified scattering and construct modified wave operators for small initial and final data respectively (1 ≤ d ≤ 4). The key novelty comes from the fact that the modified asymptotic dynamics ..."
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Abstract. We consider the cubic nonlinear Schrödinger equation posed on the spatial domain R × Td. We prove modified scattering and construct modified wave operators for small initial and final data respectively (1 ≤ d ≤ 4). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by the resonant system of this equation, which sustains interesting dynamics when d ≥ 2. As a consequence, we obtain global strong solutions (for d ≥ 2) with infinitely growing high Sobolev norms Hs. 1.