2016/01/01 by Michael Plum, Plum, Michael, Wolfgang Reichel +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Primary: 35L71 #Secondary: 34C25
paper · pdf · doi:10.48550/arxiv.1610.09203
openalex publication_date 2016/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the semilinear curl-curl wave equation s(x) \∂t2 U\n+\∇\×\∇\× U + q(x) U \± V(x) |U|p-1 U = 0 for \n(x,t)\∈ \ℝ3\×\ℝ. For any p>1 we prove the existence of\ntime-periodic spatially localized real-valued solutions (breathers) both for\nthe + and the - case under slightly different hypotheses. Our solutions are\nclassical solutions that are radially symmetric in space and decay\nexponentially to 0 as |x|\→ \∞. Our method is based on the fact that\ngradient fields of radially symmetric functions are annihilated by the\ncurl-curl operator. Consequently, the semilinear wave equation is reduced to an\nODE with r=|x| as a parameter. This ODE can be efficiently analyzed in phase\nspace. As a side effect of our analysis, we obtain not only one but a full\ncontinuum of phase-shifted breathers U(x,t+a(x)), where U is a particular\nbreather and a:\ℝ3\→\ℝ an arbitrary radially symmetric\nC2-function.\n