Nonlinear standing and rotating waves on the sphere

Abstract

We establish the existence of time-periodic solutions of semi-linear wave equations on the unit sphere in R3. The problem has been studied previously in (1982, V. Benci and Fortunato, Ann. Mat. Pura Appl.132, 215-242) using variational techniques. Our results here are much sharper: We employ delicate methods of bifurcation theory (1979, H. Kielhöfer, J. Math. Anal. Appl.68, 408-420; 1987, H. Kielhöfer and P. Kötzner, J. Appl. Math. Phys.38, 201-212) combined with well-known group-theoretic ideas to find branches of small-amplitude solutions. Precise spatio-temporal patterns of solutions are uncovered as a by-product of the analysis. Moreover, in certain cases we prove the existence of global solution branches (in the sense of P. Rabinowitz). © 2000 Academic Press.