Classification of the radial solutions of the focusing, energy-critical wave equation

Duyckaerts, Thomas; Kenig, Carlos E.; Merle, Frank

doi:10.4310/cjm.2013.v1.n1.a3

Cited by 184 publications

(324 citation statements)

References 40 publications

(75 reference statements)

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“…In this paper we presented for the first time a comprehensive, consistent overview of a general method to explicitly evaluate the large-time asymptotic solution in classical integrable systems that support this kind of reduction. We are not aware of any similar method for other integrable nonlinear models, the rather different soliton resolution conjecture [82] being the closest analog we were able to identify.…”

Section: Discussionmentioning

confidence: 99%

Quantum quench phase diagrams of ans-wave BCS-BEC condensate

et al. 2015

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We study the dynamic response of an s-wave BCS-BEC (atomic-molecular) condensate to detuning quenches within the two-channel model beyond the weak-coupling BCS limit. At long times after the quench, the condensate ends up in one of three main asymptotic states (nonequilibrium phases), which are qualitatively similar to those in other fermionic condensates defined by a global complex order parameter. In phase I the amplitude of the order parameter vanishes as a power law, in phase II it goes to a nonzero constant, and in phase III it oscillates persistently. We construct exact quench phase diagrams that predict the asymptotic state (including the many-body wave function) depending on the initial and final detunings and on the Feshbach resonance width. Outside of the weak-coupling regime, both the mechanism and the time dependence of the relaxation of the amplitude of the order parameter in phases I and II are modified. Also, quenches from arbitrarily weak initial to sufficiently strong final coupling do not produce persistent oscillations in contrast to the behavior in the BCS regime. The most remarkable feature of coherent condensate dynamics in various fermion superfluids is an effective reduction in the number of dynamic degrees of freedom as the evolution time goes to infinity. As a result, the long-time dynamics can be fully described in terms of just a few new collective dynamical variables governed by the same Hamiltonian only with "renormalized" parameters. Combining this feature with the integrability of the underlying (e.g., the two-channel) model, we develop and consistently present a general method that explicitly obtains the exact asymptotic state of the system.

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Section: Discussionmentioning

confidence: 99%

Quantum quench phase diagrams of ans-wave BCS-BEC condensate

et al. 2015

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“…[42] for a proof in the case of the KdV equation. We refer to recent works of Duyckaerts, Kenig and Merle [16,15], and references therein for general soliton decomposition results in the nonintegrable situation of the energy critical wave equation.…”

Section: Theoremmentioning

confidence: 99%

Multi-travelling waves for the nonlinear Klein-Gordon equation

Côte¹,

Martel²

2018

Trans. Amer. Math. Soc.

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“…After proving a suitable small data/pertubative theory, and carrying out the concentration compactness procedure, one reduces the proof of Theorem 1.1 to a rigidity argument, where the goal is to show that any solution to (1.1) with a pre-compact trajectory in the energy space must be a harmonic map. To prove this we use a version of the 'channels of energy' argument introduced by Duyckaerts, Kenig, and Merle in [7,8]. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d = 2 + 3 where is the equivariance class.…”

Section: Iii-1mentioning

confidence: 99%

“…However, such dynamical identities are extremely sensitive to the precise structure of the particular nonlinear wave equation under consideration, and do not easily generalize more complicated nonlinearities. An important breakthrough was made by Duyckaerts, Kenig, and Merle, [4,6,5,7,8], who developed an alternative approach called the 'channels of energy' method for rigidity arguments.…”

Section: Channels Of Energymentioning

confidence: 99%

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Stable soliton resolution for equivariant wave maps exterior to a ball

Lawrie¹

2015

Séminaire Laurent Schwartz — EDP Et Applications

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Stable soliton resolution for equivariant wave maps exterior to a ballSéminaire Laurent Schwartz -EDP et applications (2014-2015 STABLE SOLITON RESOLUTION FOR EQUIVARIANT WAVE MAPS EXTERIOR TO A BALL ANDREW LAWRIEAbstract. In this report we review the proof of the stable soliton resolution conjecture for equivariant wave maps exterior to a ball in R 3 and taking values in the 3-sphere. This is joint work with Carlos Kenig, Baoping Liu, and Wilhelm Schlag.

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Classification of the radial solutions of the focusing, energy-critical wave equation

Abstract: In this paper, we describe the asymptotic behaviour of globally defined solutions and of bounded solutions blowing up in finite time of the radial energy-critical focusing non-linear wave equation in three space dimension.

Cited by 184 publications

References 40 publications

Quantum quench phase diagrams of ans-wave BCS-BEC condensate

Quantum quench phase diagrams of ans-wave BCS-BEC condensate

Multi-travelling waves for the nonlinear Klein-Gordon equation

Stable soliton resolution for equivariant wave maps exterior to a ball

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