Breather solutions for a radially symmetric curl-curl wave equation with double power nonlinearity

Abstract

This paper is concerned with breather solutions of a radially symmetric curl-curl wave equation with double power nonlinearity ρ(x)utt+∇×(M(x)∇×u)+μ(x)u+vp(x)|u|p−1u+vq(x)|u|q−1u=0,where (x,t)∈R3×R, u:R3×R→R3 is the unknown function, M:R3→R3×3 and ρ,μ,vp,vq:R3→(0,+∞) are radially symmetric coefficient functions with 1<p<q. By considering the solutions with a special form u=y(|x|,t)x|x|, we obtain a family of ordinary differential equations (ODEs) parameterized by the radial variable r=|x|. Then we characterize periodic behaviors and analyze the joint effects of the double power nonlinear terms on the minimal period and the maximal amplitude. Under certain conditions, we construct a 2πρ(0)/μ(0)-periodic breather solution for the original curl-curl wave equation and find such a solution which can generate a continuum of phase-shifted breathers.