Solitons on pseudo-Riemannian manifolds: stability and motion

Author:
David M. A. Stuart

Journal:
Electron. Res. Announc. Amer. Math. Soc. **6** (2000), 75-89

MSC (2000):
Primary 58J45, 37K45; Secondary 35Q75, 83C10, 37K40

DOI:
https://doi.org/10.1090/S1079-6762-00-00084-6

Published electronically:
October 5, 2000

MathSciNet review:
1783091

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This is an announcement of results concerning a class of solitary wave solutions to semilinear wave equations. The solitary waves studied are solutions of the form $\phi (t,x)=e^{i\omega t}f_\omega (x)$ to semilinear wave equations such as $\Box \phi +m^2\phi =\beta (|\phi |)\phi$ on $\mathbb {R}^{1+n}$ and are called nontopological solitons. The first preprint provides a new modulational approach to proving the stability of nontopological solitons. This technique, which makes strong use of the inherent symplectic structure, provides explicit information on the time evolution of the various parameters of the soliton. In the second preprint a pseudo-Riemannian structure $\underline {g}$ is introduced onto $\mathbb {R}^{1+n}$ and the corresponding wave equation is studied. It is shown that under the rescaling $\underline {g}\to \epsilon ^{-2} \underline {g}$, with $\epsilon \to 0$, it is possible to construct solutions representing nontopological solitons concentrated along a time-like geodesic.

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Additional Information

**David M. A. Stuart**

Affiliation:
Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 OWA, UK

Email:
D.M.A.Stuart@damtp.cam.ac.uk

Keywords:
Wave equations on manifolds,
nontopological solitons,
stability,
solitary waves.

Received by editor(s):
April 30, 2000

Published electronically:
October 5, 2000

Additional Notes:
The author acknowledges support from EPSRC Grant AF/98/2492.

Communicated by:
Michael Taylor

Article copyright:
© Copyright 2000
American Mathematical Society