Existence and Newtonian limit of nonlinear bound states in the Einstein–Dirac system

Stuart, David

doi:10.1063/1.3294085

Cited by 31 publications

(16 citation statements)

References 23 publications

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“…Further, Fröhlich together with Tsai and Yau obtained similar results for nonlinear Hartree equations [132], and with Gustafson, Jonsson and Sigal, for nonlinear Schrödinger equations [133]. Stuart, Demulini and Long have proved similar results for nonlinear Einstein-Dirac, Chern-Simons-Schrödinger and Klein-Gordon-Maxwell systems [134]- [136]. Recently, Bach, Chen, Faupin, Fröhlich and Sigal proved the adiabatic effective dynamics for one electron in second-quantized Maxwell field in the presence of a slowly varying external potential [137].…”

Section: Introductionsupporting

confidence: 55%

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Attractors of Hamilton nonlinear PDEs

Komech¹

2016

DCDS-A

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This is a survey of results on long time behavior and attractors for nonlinear Hamiltonian partial differential equations, considering the global attraction to stationary states, stationary orbits, and solitons, the adiabatic effective dynamics of the solitons, and the asymptotic stability of the solitary manifolds. The corresponding numerical results and relations to quantum postulates are considered.This theory differs significantly from the theory of attractors of dissipative systems where the attraction to stationary states is due to an energy dissipation caused by a friction. For the Hamilton equations the friction and energy dissipation are absent, and the attraction is caused by radiation which brings the energy irrevocably to infinity.All the above-mentioned results [21]-[27] on the local energy decay (1.3) mean that the corresponding local attractor of small initial states consists of the zero point only. First results on the global attractors for nonlinear Hamiltonian PDEs were obtained by the author in the 1991-1995's for 1D models [37,38,39], and were later extended to nD equations. The main difficulty here is due to the absence of energy dissipation for the Hamilton equations. For example, the attraction to a (proper) attractor is impossible for any finite-dimensional Hamilton system because of the energy conservation. The problem is attacked by analyzing the energy radiation to infinity, which plays the role of dissipation. The progress relies on a novel application of subtle methods of harmonic analysis: the Wiener Tauberian theorem, the Titchmarsh convolution theorem, theory of quasi-measures, the Paley-Wiener estimates, the eigenfunction expansions for nonselfadjoint Hamilton operators based on M.G. Krein theory of J-selfadjoint operators, and others.The results obtained so far indicate a certain dependence of long-teme asymptotics of solutions on the symmetry group of the equation: for example, it may be the trivial group G = {e}, or the unitary group G = U(1), or the group of translations G = R n . Namely, the corresponding results suggest that for 'generic' autonomous equations with a Lie symmetry group G, any finite energy solution admits the asymptotics ψ(x,t) ∼ e g ± t ψ ± (x), t → ±∞.(1.4) Recently, the global attraction to stationary orbits was established for discrete in space and time nonlinear Hamilton equations [105]. The proofs required a refined version of the Titchmarsh convolution theorem for distributions on the circle [106].The main ideas of the proofs [98]-[105] rely on the radiation mechanism caused by dispersion radiation and nonlinear inflation of spectrum (Section 3.8).

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Section: Introductionsupporting

confidence: 55%

“…Further, the asymptotics of type (5.5), (5.6) were obtained in [132,133] for the nonlinear Hartree and Schrödinger equations with slowly varying external potentials, and in [134]- [136], for nonlinear Einstein-Dirac, Chern-Simon-Schrödinger and Klein-Gordon-Maxwell equations with small external fields.…”

Section: Generalizations and The Mass-energy Equivalencementioning

confidence: 94%

Attractors of Hamilton nonlinear PDEs

Komech¹

2016

DCDS-A

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“…This nondegeneracy result is an important property which is useful in implicit function type arguments. Uniqueness and nondegeneracy were originally used in [7] to study a pseudo-relativistic model, and then in [5,11,18,20,4,19] for other models. The purpose of this paper is to study the case of anisotropic media, for which the corresponding potential is…”

Section: Introductionmentioning

confidence: 99%

On uniqueness and non-degeneracy of anisotropic polarons

Ricaud

2016

Nonlinearity

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We study the anisotropic Choquard-Pekar equation which describes a polaron in an anisotropic medium. We prove the uniqueness and non-degeneracy of minimizers in a weakly anisotropic medium. In addition, for a wide range of anisotropic media, we derive the symmetry properties of minimizers and prove that the kernel of the associated linearized operator is reduced, apart from three functions coming from the translation invariance, to the kernel on the subspace of functions that are even in each of the three principal directions of the medium.

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“…A perturbation method for the construction of solitary waves in the nonlinear Dirac equation was used in [Oun00]. This work was later followed in [Gua08,CGG14] and also generalized to the Einstein-Dirac and Einstein-Dirac-Maxwell systems [RN10a,Stu10,RN10b] and to the Dirac-Maxwell system [CS12]. Our aim here is to make the perturbative approach of the seminal work [Oun00] rigorous for the important case of lower order nonlinearities.…”

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confidence: 99%

Nonrelativistic Asymptotics of Solitary Waves in the Dirac Equation with Soler-Type Nonlinearity

Boussaïd¹,

Comech²

2017

SIAM J. Math. Anal.

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Abstract. We use the perturbation theory to build solitary wave solutions φω(x)e −iωt to the nonlinear Dirac equation in R n , n ≥ 1, with the Soler-type nonlinear term f (ψ * βψ)βψ, with f (τ ) = |τ | k +o(|τ | k ), k > 0, which is continuous but not necessarily differentiable. We obtain the asymptotics of solitary waves in the nonrelativistic limit ω m; these asymptotics are important for the linear stability analysis of solitary wave solutions. We also show that in the case when the power of the nonlinearity is Schrödinger charge-critical (k = 2/n), then one has Q ′ (ω) < 0 for ω m, with Q(ω) being the charge of the corresponding solitary wave; this implies the absence of the degeneracy of zero eigenvalue of the linearization at this solitary wave. In the present analysis, we use the bifurcation approach to construct solitary wave solutions to the nonlinear Dirac equation with scalar-type self-interaction, known as the Soler model [Iva38, Sol70]:

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Existence and Newtonian limit of nonlinear bound states in the Einstein–Dirac system

Cited by 31 publications

References 23 publications

Attractors of Hamilton nonlinear PDEs

Attractors of Hamilton nonlinear PDEs

On uniqueness and non-degeneracy of anisotropic polarons

Nonrelativistic Asymptotics of Solitary Waves in the Dirac Equation with Soler-Type Nonlinearity

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