A Breather Construction for a Semilinear Curl-Curl Wave Equation with Radially Symmetric Coefficients

We consider the semilinear curl-curl wave equation s(x)∂t2U+∇×∇×+q(x)U±V(x)|U|p− 1U=0for(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 C2-function. © 2016, Orthogonal Publishing and Springer International Publishing.