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

Abstract

We consider the semilinear curl-curl wave equation sFor any p > 1 we prove the existence of timeperiodic 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 : R 3 → R an arbitrary radially symmetric C 2 -function.

Construction of breather solutions for nonlinear Klein-Gordon equations on periodic metric graphs

“…In both cases a spectral gap near zero of the wave operator acting on time-periodic functions with a given time-period is vital for the results. Recently, in [23] another existence result for vector-valued breathers for a 3 + 1-dimensional semilinear curl-curl wave equation with radial symmetry appeared. It is based on ODE-methods and the fact that the breather can be found as a gradient of a spatially radially symmetric function.…”

Variational methods for breather solutions of nonlinear wave equations

“…The first deals with a semilinear curl-curl wave equation in R 3 × R where −u xx is replaced by ∇ × ∇ × u in (1) and u is a three-dimensional vector field on R 3 . Using that this part in the equation actually vanishes for gradient fields, Plum and Reichel [28] succeed in proving the existence of exponentially localized breather solutions via ODE methods for suitable radially symmetric coefficient functions s, q and power-type nonlinearities f . As far as we know, this is the only result dealing with strongly localized breathers in higher dimensions, i.e., U(t, •) ∈ L 2 (R N ) for almost all t ∈ R. Recently, the second author suggested a new construction of (even in time) breathers [29] for the cubic Klein-Gordon equation that we will refer to as weakly localized in space.…”

A uniqueness result for the Sine-Gordon breather

“…(2) satisfying uðÁ; tÞ 2 L 2 ðR N Þ for almost all t 2 R and N ! 2, see however [11] for a an existence result for semilinear curl-curl equations for N ¼ 3. In the case N ¼ 1 strongly localized breather solutions different from the sine-Gordon breather have been found for nonlinear wave equations of the form…”