Dynamic of Threshold Solutions for Energy-Critical NlS

Duyckaerts, Thomas; Merle, Frank

doi:10.1007/s00039-009-0707-x

Cited by 121 publications

(217 citation statements)

References 20 publications

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“…The proofs of Theorems 2 and 3 will show that the behavior of the solutions of (1.1) at the threshold is very close to the one of the energy-critical equation described in the radial case in [8]. In particular, in both cases, the existence of the special solutions Q ± (W ± in the energy-critical case) derives from the existence of two real nonzero eigenvalues for the linearized operator around the periodic solution e it Q (respectively around the stationary solution W ).…”

Section: Remark It Is Worth Linking Theḣmentioning

confidence: 78%

“…Note that results in this paper are more complete than those in [8], which are restricted to radial solutions 3 and the blow-up of the special solution W + is shown only in the space dimension 5. This is due to the fact that we have more freedom in our setting than in the energy-critical one: the set of solutions of (1.1) is stable by the Galilean transformation and the ground state Q decays exponentially at infinity.…”

Section: Remark It Is Worth Linking Theḣmentioning

confidence: 80%

“…The proof of Proposition 3.1 is similar to the one of Proposition 6.1 in [8]. We start with the construction of a family of approximate solutions to (1.1) that satisfy (3.1), and then prove the existence of U A by a fixed point argument around an approximate solution.…”

Section: Remark 33mentioning

confidence: 91%

“…We refer to [8,Subsection 3.3] and [9, Subsection 3.3] for similar arguments. The two ingredients of the proof of Proposition 6.1 are the localized virial argument (Lemma 6.7) and a precise control of the variations of the parameter x(t) (Lemma 6.8).…”

Section: Exponential Convergencementioning

confidence: 99%

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Threshold solutions for the focusing 3D cubic Schrödinger equation

Duyckaerts¹,

Roudenko²

2010

Rev. Mat. Iberoam.

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143

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We study the focusing 3d cubic NLS equation with H 1 data at the mass-energy threshold, namely, when. In earlier works of Holmer-Roudenko and Duyckaerts-Holmer-Roudenko, the behavior of solutions (i.e., scattering and blow up in finite time) was classified when. In this paper, we first exhibit 3 special solutions: e it Q and Q ± , where Q is the ground state, Q ± exponentially approach the ground state solution in the positive time direction, Q + has finite time blow up and Q − scatters in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up toḢ 1/2 symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the mass-energy threshold. These results are obtained by studying the spectral properties of the linearized Schrödinger operator in this mass-supercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.

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Section: Remark It Is Worth Linking Theḣmentioning

confidence: 78%

Section: Remark It Is Worth Linking Theḣmentioning

confidence: 80%

Section: Remark 33mentioning

confidence: 91%

Section: Exponential Convergencementioning

confidence: 99%

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Threshold solutions for the focusing 3D cubic Schrödinger equation

Duyckaerts¹,

Roudenko²

2010

Rev. Mat. Iberoam.

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143

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“…The following theorem shown in [KM06a] for the case E(u 0 ) < E(W ) and in [DM07a] for the case E(u 0 ) = E(W ), is the analoguous of Theorem A for equation (5.1).…”

Section: Estimate Of the Scattering Norm For Energy-critical Focusingmentioning

confidence: 82%

Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

Duyckaerts¹,

Merle²

2009

Indiana Univ. Math. J.

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Abstract. We consider the energy-critical semilinear focusing wave equation in dimension N = 3, 4, 5. An explicit solution W of this equation is known. By the work of C. Kenig and F. Merle, any solution of initial condition (u0, u1) such that E(u0, u1) < E(W, 0) and ∇u0 L 2 < ∇W L 2 is defined globally and has finite L 2(N +1) N −2 t,x -norm, which implies that it scatters. In this note, we show that the supremum of the L 2(N +1) N −2 t,x -norm taken on all scattering solutions at a certain level of energy below E(W, 0) blows-up logarithmically as this level approaches the critical value E(W, 0). We also give a similar result in the case of the radial energy-critical focusing semilinear Schrödinger equation. The proofs rely on the compactness argument of C. Kenig and F. Merle, on a classification result, due to the authors, at the energy level E(W, 0), and on the analysis of the linearized equation around W .

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A critical center‐stable manifold for Schrödinger's equation in three dimensions

Beceanu

2012

Comm Pure Appl Math

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Consider the focusing P H 1=2 -critical semilinear Schrödinger equation in R 3 (0.1)It admits an eight-dimensional manifold of special solutions called ground state solitons.We exhibit a codimension-1 critical real analytic manifold N of asymptotically stable solutions of (0.1) in a neighborhood of the soliton manifold. We then show that N is center-stable, in the dynamical systems sense of Bates and Jones, and globally-in-time invariant.Solutions in N are asymptotically stable and separate into two asymptotically free parts that decouple in the limit-a soliton and radiation. Conversely, in a general setting, any solution that stays P H 1=2 -close to the soliton manifold for all time is in N .The proof uses the method of modulation. New elements include a different linearization and an endpoint Strichartz estimate for the time-dependent linearized equation.The proof also uses the fact that the linearized Hamiltonian has no nonzero real eigenvalues or resonances. This has recently been established in the case treated here-of the focusing cubic NLS in R 3 -by the work of Marzuola and Simpson and Costin, Huang, and Schlag.

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Dynamic of Threshold Solutions for Energy-Critical NlS

Cited by 121 publications

References 20 publications

Threshold solutions for the focusing 3D cubic Schrödinger equation

Threshold solutions for the focusing 3D cubic Schrödinger equation

Scattering norm estimate near the threshold for energy-critical focusing semilinear wave equation

A critical center‐stable manifold for Schrödinger's equation in three dimensions

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