Nonlinear Standing and Rotating Waves on the Sphere

We establish the existence of time-periodic solutions of semi-linear wave equations on the unit sphere in 3. 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).