A breather construction for a semilinear curl-curlwave equation with radially symmetric coefficients

Abstract. We consider the semilinear curl-curl wave equation s(x)∂2 U + ∇ × ∇ × U + q(x)U ± V (x)|U |p−1 U = 0 for (x, t) ∈ R3 × R. For any p > 1 we prove the existence of time- periodic spatially localized real-valued solutions (breathers) both for the + and the − case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to 0 as |x| → ∞. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curl-curl operator. Consequently, the semilinear wave equation is reduced to an ODE with r = |x| as a parameter. This ODE can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain not only one but a full continuum of phase-shifted breathers U (x, t+a(x)), where U is a particular breather and a : R3 → R an arbitrary radially symmetric C 2 -function.