Breathers and rogue waves for semilinear curl-curl wave equations

Abstract

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