Scattering for the non-radial 3D cubic nonlinear Schrödinger equation

Duyckaerts, Thomas; Holmer, Justin; Roudenko, Svetlana

doi:10.4310/mrl.2008.v15.n6.a13

Cited by 247 publications

(357 citation statements)

References 30 publications

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“…Then the classification by [1] implies that u ∞ is, modulo time translation, either the soliton e −it Q itself or the unique solution w + which is exponentially converging to e −it Q as t → −∞ and scattering to 0 as t → +∞. The strong convergence at t = 0 implies d ∞ ( u ∞ (0)) = δ X > 0, precluding the soliton case.…”

Section: 22mentioning

confidence: 99%

“…All the solutions below the excited states E(u) < E 1 (M(u)) are completely split into (1) and (2) with the same behavior in t > 0 and in t < 0, which is explicitly predictable by the initial data, using the virial functional: The difference between below and above the excited energy are the new type (3), and solutions with different types of behavior in t > 0 and in t < 0, namely transition among (1)-(3).…”

Section: Types Of Behaviormentioning

confidence: 99%

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Global Dynamics Above the First Excited Energy for the Nonlinear Schrödinger Equation with a Potential

Nakanishi

2017

Commun. Math. Phys.

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Abstract. Consider the focusing nonlinear Schrödinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and solitons with positive large energy, which are unstable. We classify the global dynamics into 9 sets of solutions in the phase space including both solitons, restricted by small mass, radial symmetry, and an energy bound slightly above the second lowest one of solitons. The classification includes a stable set of solutions which start near the first excited solitons, approach the ground states locally in space for large time with large radiation to the spatial infinity, and blow up in negative finite time.

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Section: 22mentioning

confidence: 99%

Section: Types Of Behaviormentioning

confidence: 99%

Global Dynamics Above the First Excited Energy for the Nonlinear Schrödinger Equation with a Potential

Nakanishi

2017

Commun. Math. Phys.

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“…It is sufficient to show that for every time-sequence τ n ≥ 0, there exists (extracting if necessary) a subsequence x n such that u(x + x n , τ n ) has a limit in H 1 (see e.g. [7,Appendix] …”

Section: Which Does Not Scatter For Positive Times Then There Existsmentioning

confidence: 99%

“…Let u be a solution of (1.1). Applying to u, as in [7,Section 4], the Galilean transformation with parameter…”

Section: Remark It Is Worth Linking Theḣmentioning

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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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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Scattering for the non-radial 3D cubic nonlinear Schrödinger equation

Cited by 247 publications

References 30 publications

Global Dynamics Above the First Excited Energy for the Nonlinear Schrödinger Equation with a Potential

Global Dynamics Above the First Excited Energy for the Nonlinear Schrödinger Equation with a Potential

Threshold solutions for the focusing 3D cubic Schrödinger equation

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

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