Threshold solutions for the focusing 3D cubic Schrödinger equation

Duyckaerts, Thomas; Roudenko, Svetlana

doi:10.4171/rmi/592

Cited by 91 publications

(155 citation statements)

References 35 publications

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“…Setting and the main result. As a first step toward the above problem, we consider the NLS with a potential iu + Hu = s|u| 2 u, H := −∆ + V, s = ±, 7) in the simplest non-trivial setting, namely the case with the unique eigenvalue…”

Section: 2mentioning

confidence: 99%

Global dynamics below excited solitons for the nonlinear Schrödinger equation with a potential

Nakanishi¹

2017

J. Math. Soc. Japan

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Consider the nonlinear Schrödinger equation (NLS) with a potential with a single negative eigenvalue. It has solitons with negative small energy, which are asymptotically stable, and, if the nonlinearity is focusing, then also solitons with positive large energy, which are unstable. In this paper we classify the global dynamics below the second lowest energy of solitons under small mass and radial symmetry constraints.2010 Mathematics Subject Classification. 35Q55.

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Section: 2mentioning

confidence: 99%

Global dynamics below excited solitons for the nonlinear Schrödinger equation with a potential

Nakanishi¹

2017

J. Math. Soc. Japan

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“…By studying the dichotomy of the radial solution of with the energy below the threshold, we can show that it is just the energy of the ground state W for . For the applications of the concentration‐compactness in the scattering theory and rigidity theory of the critical NLS, NLW, NLKG, and Hartree equations, please see .…”

Section: Introductionmentioning

confidence: 99%

“…By studying the dichotomy of the radial solution of (1.1) with the energy below the threshold, we can show that it is just the energy of the ground state W for (1.6). For the applications of the concentration-compactness in the scattering theory and rigidity theory of the critical NLS, NLW, NLKG, and Hartree equations, please see [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. Now, for ' 2 H 1 , we denote the two-parameter scaling quantity ' 3, 2 by ' 3, 2 .x/ D e 3 '.e 2 x/.…”

Section: Introductionmentioning

confidence: 99%

The dynamics of the 3D NLS with the convolution term

Xia

2015

Math Methods in App Sciences

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We consider the nonlinear Schrödinger equation (NLSH) with the convolution combined term iut+normalΔu=−|u|4u+(|x|−γ*|u|2)u,1em2<γ<3 (CNLSH) in the energy space H1(double-struckR3). We firstly use a variational approach to give a dichotomy of scattering and blow up for the radial solution with the energy below the threshold, which is given by the ground state W for the energy‐critical NLS: iut+Δu=−|u|4u. The basic strategy is the concentration‐compactness arguments from Kenig and Merle. We overcome the main difficulties coming from the lack of scaling invariance and the non‐local property of the convolution term. Our result shows that the focusing, trueḢ1‐critical term −|u|4u plays the decisive role of the threshold of the scattering solution of (CNLSH) in the energy space. Copyright © 2015 John Wiley & Sons, Ltd.

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“…Following this approach, Holmer and Roudenko [33], Duyckaerts, Holmer, and Roudenko [23], and Duyckaerts and Roudenko [25] established corresponding results for the P H 1=2 -critical equation (1.1). Their main findings may be summarized as follows: THEOREM 1.2.…”

Section: Orbital and Asymptotic Stabilitymentioning

confidence: 99%

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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Threshold solutions for the focusing 3D cubic Schrödinger equation

Cited by 91 publications

References 35 publications

Global dynamics below excited solitons for the nonlinear Schrödinger equation with a potential

Global dynamics below excited solitons for the nonlinear Schrödinger equation with a potential

The dynamics of the 3D NLS with the convolution term

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

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