Solitons on pseudo-Riemannian manifolds: stability and motion

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.