Multisolitons for the Defocusing Energy Critical Wave Equation with Potentials

Chen, Gong

doi:10.1007/s00220-018-3170-4

Cited by 10 publications

(5 citation statements)

References 13 publications

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“…To handle the strong interactions of solitons and potentials, in this subsection we establish some reversed type local decay estimates. These estimates are also important to handle multisoliton structures as in [GC4].…”

Section: Note Thatmentioning

confidence: 99%

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Strichartz Estimates for Wave equations with Charge Transfer Hamiltonian

Chen

2016

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We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer Hamiltonian in R 3 :The energy estimate and the local energy decay of a scattering state are also established. In order to study nonlinear multisoltion systems, we will present the inhomogeneous generalizations of Strichartz estimates and local decay estimates. As an application of our results, we show that scattering states indeed scatter to solutions to the free wave equation. These estimates for this linear models are also of crucial importance for problems related to interactions of potentials and solitons, for example, in [GC4].

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Section: Note Thatmentioning

confidence: 99%

“…in this paper, we prove Strichartz estimates, energy estimates, the local energy decay which are essential to analyze the stability of multi-soliton states. In Chen [GC4], relying on this linear model, we construct a multisoliton structure to the defocusing energy critical wave equation with potentials in R 3 :…”

Section: Introductionmentioning

confidence: 99%

Strichartz Estimates for Wave equations with Charge Transfer Hamiltonian

Chen

2016

Preprint

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“…A different method of construction of multi-solitons, based on the fixed point argument of Merle in [54], has been introduced by Le Coz, Li and Tsai in [41] for the NLS equation. This strategy has also been used by Chen for the wave equation [9] and by Van Tin for the derivative NLS [67].…”

Section: Solitary Wave Solutionsmentioning

confidence: 99%

Asymptotic $N$-soliton-like solutions of the fractional Korteweg-de Vries equation

Eychenne¹

2021

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We construct N -soliton solutions for the fractional Korteweg-de Vries (fKdV) equationin the whole sub-critical range α ∈] 1 2 , 2[. More precisely, if Qc denotes the ground state solution associated to fKdV evolving with velocity c, then given 0where ρ ′ j (t) ∼ c j as t → +∞. The proof adapts the construction of Martel in the generalized KdV setting [Amer. J. Math. 127 (2005), pp. 1103-1140]) to the fractional case. The main new difficulties are the polynomial decay of the ground state Qc and the use of local techniques (monotonicity properties for a portion of the mass and the energy) for a non-local equation. To bypass these difficulties, we use symmetric and non-symmetric weighted commutator estimates. The symmetric ones were proved by Kenig, Martel and Robbiano [Annales de l'IHP Analyse Non Linéaire 28 (2011), pp. 853-887], while the non-symmetric ones seem to be new.

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“…In R 3 , it is known that the endpoint Strichartz estimate L 2 t L ∞ x fails both for the wave and Klein-Gordon equations, see for example Machihara-Nakamura-Nakanishi-Ozawa [MNNO]. But in many nonlinear stability analysis problems, for example in Chen [Che2] and Jia-Liu-Schlag-Xu [JLSchX], one has to deal with the quadratic terms. Then the reversed type estimates L ∞…”

mentioning

confidence: 99%

“…Except for the homogeneous Strichartz estimate, the standard Strichartz estimates are not Lorentz invariant. But using the reversed type estimates, one can obtain some Lorentz invariance using a family of reversed type estimates which will be important in the analysis of multisoliton, see Chen [Che1,Che2]. We should point out that these reversed type estimates for perturbed equations do not follow from the L p boundedness of wave operators.…”

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confidence: 99%

Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$

Beceanu,

Chen

2021

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In this paper we prove standard and reversed Strichartz estimates for the Klein-Gordon equation in R 3+1 . Instead of the Fourier theory, our analysis is based on fundamental solutions of the free equations and fractional integrations. In the final part of this paper, we apply Strichartz estimates in the study of a semilinear Klein-Gordon equation. Contents 1. Introduction 1.1. The linear Klein-Gordon equation 1.2. The free equation 1.3. The perturbed equation 1.4. Main results 1.5. The semilinear Klein-Gordon equation 1.6. Discussion of the results 1.7. Notations 1.8. Organization 2. The free Klein-Gordon kernels 2.1. The cosine kernel 2.2. Fractional integration 2.3. Klein-Gordon sine propagator bounds 2.4. Fractional powers 2.5. The free sine propagator 2.6. Pointwise decay estimates 3. The Klein-Gordon equation with a scalar potential 3.1. Standard Strichartz estimates 4. Nonlinear applications Appendix A. Bessel and Hankel functions Appendix B. Sharp Agmon estimates Appendix C. Comparison of Sobolev spaces References

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Multisolitons for the Defocusing Energy Critical Wave Equation with Potentials

Cited by 10 publications

References 13 publications

Strichartz Estimates for Wave equations with Charge Transfer Hamiltonian

Strichartz Estimates for Wave equations with Charge Transfer Hamiltonian

Asymptotic $N$-soliton-like solutions of the fractional Korteweg-de Vries equation

Strichartz estimates for the Klein--Gordon equation in $\mathbb{R}^{3+1}$

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