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

Abstract

We consider the semilinear curl-curl wave equation $$s\left( x \right)\partial _t^2U + \nabla \times \nabla \times + q\left( x \right)U \pm V\left( x \right){\left| U \right|^{p - {\text{ 1}}}}U = 0for\left( {x,t} \right) \in {\mathbb{R}^3} \times \mathbb{R}.$$ 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.