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We consider the semilinear curl-curl wave equation $$sleft( x right)partial _t^2U + nabla times nabla times + qleft( x right)U pm Vleft( x right){left| U right|^{p - {text{ 1}}}}U = 0forleft( {x,t} right)  {mathbb{R}^3} times mathbb{R}.$$For any p u003c 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.

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