Dynamic of Threshold Solutions for Energy-Critical NlS

By Thomas Duyckaerts, F. Merle. 2007

161 citations recorded by the source corpus.

Abstract.We consider the energy-critical non-linear focusing Schrödinger equation in dimension N = 3, 4, 5. An explicit stationary solution, W, of this equation is known. In [KeM], the energy E(W) has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present article, we study the dynamics at the critical level E(u) = E(W) and classify the corresponding solutions. This gives in particular a dynamical characterization of W.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9dfc3a937d1c4a312f462bbed11d6eae1a794252

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In Control and Systems Engineering

1. F. Merle
2. Pierre Raphael
3. Yvan Martel
4. Jérémie Szeftel
5. Vianney Combet
6. Xiaoyi Zhang
7. Aynur Bulut
8. Dong Li

In Mathematical Physics

1. F. Merle
2. Pierre Raphael
3. Yvan Martel
4. Jérémie Szeftel
5. Carlos E. Kenig
6. Aynur Bulut
7. Vianney Combet
8. Xiaoyi Zhang

Citations in the corpus

• Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS (2010, methodology): “[23] Merle, F., Determination of minimal blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power.”
• Multi-Soliton Solutions for the Supercritical gKdV Equations (2010, result): “[14] Y. Martel and F. Merle.”
• Blow up for the critical gKdV equation II: minimal mass dynamics (2012, methodology): “Related rigidity theorems near the solitary wave were recently obtained by Nakanishi and Schlag [46], [47] for super critical wave and Schrödinger equations using the invariant set methods of Beresticky, Cazenave [2], the Kenig-Merle concentration compactness approach [14], the classification of min”
• Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions (2009, methodology): “he energy threshold E (u0,u1) = E (W,0) in high dimensions. Such dynamics were analyzed by Duyckaerts and Merle [5] for 3 ≤ d ≤ 5, and recently by Li and Zhang [20] in high dimensions d ≥ 6 (see also [6] and [21] for the NLS case). 2In Theorem 3.6 we do not assume smallness in exotic Strichartz spac”