Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation
TLDR
In this paper, the cubic defocusing nonlinear Schrodinger equa- tion on the two dimensional torus is considered and smooth solutions for which the support of the conserved energy moves to higher Fourier modes are presented.Abstract:
We consider the cubic defocusing nonlinear Schrodinger equa- tion on the two dimensional torus We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes This behavior is quantified by the growth of higher Sobolev norms: given any δ � 1 ,K � 1 ,s >1, we construct smooth initial data u0 withu0� H s K at some time T This growth occurs despite the Hamiltonian's bound onu(t)� ˙ H 1 and despite the conservation of the quantityu(t)� L2read more
Citations
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Global solutions of nonlinear schrodinger equations
TL;DR: In this paper, the existence of global solutions to the Cauchy problem of nonlinear SchrSdinger equation by establishing time weight function spaces and using the contraction mapping principle was studied.
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On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions
TL;DR: In this article, the frequencies of long-lived gravitational quasinormal modes of AdS black holes in various dimensions were calculated and a new class of asymptotically AdS solutions describing noncoalescing black hole binaries were discussed.
Journal ArticleDOI
On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions
TL;DR: In this paper, the frequencies of long-lived gravitational quasinormal modes of AdS black holes in various dimensions were calculated and a new class of asymptotically anti-de Sitter solutions describing noncoalescing black hole binaries were discussed.
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Modified scattering for the cubic Schrödinger equation on product spaces and applications
TL;DR: In this paper, the cubic nonlinear Schrodinger equation was considered on the spatial domain and modified scattering and wave operators for small initial and final data were constructed for small data.
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A KAM algorithm for the resonant non-linear Schrödinger equation
Claudio Procesi,Michela Procesi +1 more
TL;DR: In this paper, the existence of large families of stable and unstable quasi-periodic solutions for the NLS in any number of independent frequencies was proved through a KAM algorithm.
References
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Book
Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals
Elias M. Stein,Timothy S Murphy +1 more
TL;DR: In this article, the authors introduce the Heisenberg group and describe the Maximal Operators and Maximal Averages and Oscillatory Integral Integrals of the First and Second Kind.
Journal Article
Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media
Journal ArticleDOI
Stability of periodic waves of finite amplitude on the surface of a deep fluid
TL;DR: In this article, the stability of steady nonlinear waves on the surface of an infinitely deep fluid with a free surface was studied. And the authors considered the problem of stability of surface waves as part of the more general problem of nonlinear wave in media with dispersion.
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Semilinear Schrodinger Equations
TL;DR: In this article, the linear Schrodinger equation and local Cauchy problem are studied in the repulsive case and the attractive case, respectively, and the smoothing effect is considered.