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Stuart, David M. A.

Geodesics and the Einstein nonlinear wave system. (English) Zbl 1072.58021

J. Math. Pures Appl. (9) 83, No. 5, 541-587 (2004).
Summary: The system under consideration is Einstein’s equation \(R_{\mu\nu} ({\mathbf g})- g_{\mu\nu}R({\mathbf g})/2=8\pi{\mathbf G} T_{\mu\nu}\) for a pseudo-Riemannian metric \({\mathbf g}\) coupled to a semi-linear wave equation for a complex function \(\varphi\). Assume that this wave equation on Minkowski space admits a stable solitary wave of the type known as nontopological solitons. The system is studied in the scaling limit in which the solitons have small size \(\varepsilon\) and amplitude \(\delta\) with \(\delta\leq\delta_0\varepsilon^{7/4}\). It is proved that, for \(\varepsilon\) sufficiently small, given a solution of the vacuum Einstein equation, i.e., a Ricci flat pseudo-Riemannian metric \(\gamma\), there exists a finite time interval, independent of \(\varepsilon,\delta\), on which there is a solution of the full system \(({\mathbf g},\varphi)\) with \(({\mathbf g}-\gamma)\) small and \(\varphi\) close to a nontopological soliton centred on a time-like geodesic (in appropriate Sobolev norms).

Cited in 1 Review
Cited in 6 Documents

MSC:

58J45 Hyperbolic equations on manifolds
35L70 Second-order nonlinear hyperbolic equations
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)

Keywords:

Godesic; Nonlinear wave equations on manifolds; Solitons
PDFBibTeX XMLCite
\textit{D. M. A. Stuart}, J. Math. Pures Appl. (9) 83, No. 5, 541--587 (2004; Zbl 1072.58021)
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References:

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