Breathers and rogue waves for semilinear curl-curl wave equations

Abstract We consider localized solutions of variants the semilinear curl-curl wave equation $$s(x) \partial _t^2 U +\nabla \times \nabla + q(x) \pm V(x) \vert ^{p-1} = 0$$ s ( x ) ∂ t 2 U + ∇ × q ± V | p - 1 = 0 for $$(x,t)\in {\mathbb {R}}^3\times {R}}$$ , ∈ R 3 and arbitrary $$p>1$$ > . Depending on coefficients s , q V we can prove existence three types solutions: time-periodic decaying to 0 at spatial infinity, tending a nontrivial profile infinity (both are called breathers), rogue waves which converge both temporal infinity. Our weak take form gradient fields. Thus they belong kernel curl-operator so that due structural assumptions is reduced an ODE. Since space dependence in ODE just parametric analyze by phase plane techniques thus establish described above. Noteworthy side effects our analysis compact support breathers fact one solution ( x t ) already generates full continuum phase-shifted $$U(x,t+b(x))$$ b where continuous function $$b:{\mathbb {R}}^3\rightarrow : → belongs suitable admissible family.