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Solitons on pseudo-Riemannian manifolds: stability and motion
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## Solitons on pseudo-Riemannian manifolds: stability and motion

### David M. A. Stuart

**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
$\underline{g}\to\epsilon^{-2} \ulg$, with $\epsilon\to 0$, it is
possible to
construct solutions representing nontopological solitons
concentrated along a time-like geodesic.

*Copyright 2000 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**06** (2000), pp. 75-89
- Publisher Identifier: S 1079-6762(00)00084-6
- 2000
*Mathematics Subject Classification*. Primary 58J45, 37K45; Secondary 35Q75, 83C10, 37K40
*Key words and phrases*. Wave equations on manifolds, nontopological solitons, stability, solitary waves
- Received by the editors April 30, 2000
- Posted on October 5, 2000
- Communicated by Michael Taylor
- Comments (When Available)

**David M. A. Stuart**

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

*E-mail address:* `D.M.A.Stuart@damtp.cam.ac.uk`

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

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