Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq; Laurent Thomann; Nikolay Tzvetkov

Displaying similar documents to “Long time dynamics for the one dimensional non linear Schrödinger equation”

Blow-up for the focusing energy critical nonlinear Schrödinger equation with confining harmonic potential

Xing Cheng, Yanfang Gao (2014)

Colloquium Mathematicae

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The focusing nonlinear Schrödinger equation (NLS) with confining harmonic potential $i \partial_{t} u + 1 / 2 Δ u - 1 / 2 | x | ² u = - {| u |}^{4 / (d - 2)} u, x \in ℝ^{d}$ , is considered. By modifying a variational technique, we shall give a sufficient condition under which the corresponding solution blows up.

Semiclassical measures for the Schrödinger equation on the torus

Nalini Anantharaman, Fabricio Macià (2014)

Journal of the European Mathematical Society

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In this article, the structure of semiclassical measures for solutions to the linear Schrödinger equation on the torus is analysed. We show that the disintegration of such a measure on every invariant lagrangian torus is absolutely continuous with respect to the Lebesgue measure. We obtain an expression of the Radon-Nikodym derivative in terms of the sequence of initial data and show that it satisfies an explicit propagation law. As a consequence, we also prove an observability inequality,...

Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case

Jean Bourgain, Aynur Bulut (2014)

Journal of the European Mathematical Society

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We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in $ℝ^{d}$ to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in $ℝ^{3}$ . The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated...

Critical case of nonlinear Schrödinger equations with inverse-square potentials on bounded domains

Toshiyuki Suzuki (2014)

Mathematica Bohemica

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Nonlinear Schrödinger equations (NLS) $_{a}$ with strongly singular potential ${a | x |}^{- 2}$ on a bounded domain $Ω$ are considered. If $Ω = ℝ^{N}$ and $a > - {(N - 2)}^{2} / 4$ , then the global existence of weak solutions is confirmed by applying the energy methods established by N. Okazawa, T. Suzuki, T. Yokota (2012). Here $a = - {(N - 2)}^{2} / 4$ is excluded because $D (P_{a (N)}^{1 / 2})$ is not equal to $H^{1} (ℝ^{N})$ , where $P_{a (N)} : = - Δ - {(N - 2)}^{2} / {(4 | x |}^{2})$ is nonnegative and selfadjoint in $L^{2} (ℝ^{N})$ . On the other hand, if $Ω$ is a smooth and bounded domain with $0 \in Ω$ , the Hardy-Poincaré inequality is proved in J. L. Vazquez, E. Zuazua...

Trudinger–Moser inequality on the whole plane with the exact growth condition

Slim Ibrahim, Nader Masmoudi, Kenji Nakanishi (2015)

Journal of the European Mathematical Society

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Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^{\infty}$ . It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modied versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the...

Square functions associated to Schrödinger operators

I. Abu-Falahah, P. R. Stinga, J. L. Torrea (2011)

Studia Mathematica

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We characterize geometric properties of Banach spaces in terms of boundedness of square functions associated to general Schrödinger operators of the form ℒ = -Δ + V, where the nonnegative potential V satisfies a reverse Hölder inequality. The main idea is to sharpen the well known localization method introduced by Z. Shen. Our results can be regarded as alternative proofs of the boundedness in H¹, $L^{p}$ and BMO of classical ℒ-square functions.

Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity

Jean Bourgain, W. Wang (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Solitons and Gibbs Measures for Nonlinear Schrödinger Equations

K. Kirkpatrick (2012)

Mathematical Modelling of Natural Phenomena

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We review some recent results concerning Gibbs measures for nonlinear Schrödinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the Gibbs measures of the discrete NLS in three dimensions, where there is a striking phase transition to soliton-like behavior.

From bosonic grand-canonical ensembles to nonlinear Gibbs measures

Nicolas Rougerie (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

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In a recent paper, in collaboration with Mathieu Lewin and Phan Thành Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like functionals could be derived from many-body quantum mechanics, in a mean-field limit. This text summarizes these findings. It focuses on the simplest, but most physically relevant, case we could treat so far, namely that of the defocusing cubic NLS functional on a 1D interval. The measure obtained in the limit, which lives over $H^{1 / 2 - ϵ}$ , has been...

Positive solutions for nonlinear Schrödinger equations with deepening potential well

Zhengping Wang, Huan-Song Zhou (2009)

Journal of the European Mathematical Society

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Semiclassical states for weakly coupled nonlinear Schrödinger systems

Eugenio Montefusco, Benedetta Pellacci, Marco Squassina (2008)

Journal of the European Mathematical Society

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We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.

On the boundary convergence of solutions to the Hermite-Schrödinger equation

Peter Sjögren, J. L. Torrea (2010)

Colloquium Mathematicae

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In the half-space $ℝ^{d} \times ℝ ₊$ , consider the Hermite-Schrödinger equation i∂u/∂t = -Δu + |x|²u, with given boundary values on $ℝ^{d}$ . We prove a formula that links the solution of this problem to that of the classical Schrödinger equation. It shows that mixed norm estimates for the Hermite-Schrödinger equation can be obtained immediately from those known in the classical case. In one space dimension, we deduce sharp pointwise convergence results at the boundary by means of this link.

Non-generic blow-up solutions for the critical focusing NLS in 1-D

Joachim Krieger, Wilhelm Schlag (2009)

Journal of the European Mathematical Society

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We consider the $L^{2}$ -critical focusing non-linear Schrödinger equation in $1 + 1$ -d. We demonstrate the existence of a large set of initial data close to the ground state soliton resulting in the pseudo-conformal type blow-up behavior. More specifically, we prove a version of a conjecture of Perelman, establishing the existence of a codimension one stable blow-up manifold in the measurable category.

Propagation of uniform Gevrey regularity of solutions to evolution equations

Todor Gramchev, Ya-Guang Wang (2003)

Banach Center Publications

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We investigate the propagation of the uniform spatial Gevrey $G^{σ}$ , σ ≥ 1, regularity for t → +∞ of solutions to evolution equations like generalizations of the Euler equation and the semilinear Schrödinger equation with polynomial nonlinearities. The proofs are based on direct iterative arguments and nonlinear Gevrey estimates.

Dunkl-Schrödinger Equations with and without Quadratic Potentials

Mejjaoli, Hatem (2011)

Serdica Mathematical Journal

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2000 Mathematics Subject Classification: Primary 42A38. Secondary 42B10. The purpose of this paper is to study the dispersive properties of the solutions of the Dunkl-Schrödinger equation and their perturbations with potential. Furthermore, we consider a few applications of these results to the corresponding nonlinear Cauchy problems.

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x \in ℝ^{N}$ , $u \in H ¹ (ℝ^{N})$ , where λ is a positive parameter, a and b are positive functions, while $f : ℝ \to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.

A Survey on Systems of Nonlinear Schrödinger Equations

Antonio Ambrosetti (2008)

Bollettino dell'Unione Matematica Italiana

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We survey some recent results dealing with some classes of systems of nonlinear Schrödinger equations.

Time-dependent Schrödinger perturbations of transition densities

Krzysztof Bogdan, Wolfhard Hansen, Tomasz Jakubowski (2008)

Studia Mathematica

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We construct the fundamental solution of $\partial_{t} - Δ_{y} - q (t, y)$ for functions q with a certain integral space-time relative smallness, in particular for those satisfying a relative Kato condition. The resulting transition density is comparable to the Gaussian kernel in finite time, and it is even asymptotically equal to the Gaussian kernel (in small time) under the relative Kato condition. The result is generalized to arbitrary strictly positive and finite time-nonhomogeneous transition densities on measure...