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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.