Travelling waves for Maxwell’s equations in nonlinear and nonsymmetric media

Abstract

AbstractWe look for travelling wave fields $$\begin{aligned} E(x,y,z,t)&= U(x,y) \cos (kz+\omega t)+ {\widetilde{U}}(x,y)\sin (kz+\omega t),\\&\quad (x,y,z)\in \mathbb {R}^3,\, t\in \mathbb {R} \end{aligned}$$ E ( x , y , z , t ) = U ( x , y ) cos ( k z + ω t ) + U ~ ( x , y ) sin ( k z + ω t ) , ( x , y , z ) ∈ R 3 , t ∈ R satisfying Maxwell’s equations in a nonlinear medium which is not necessarily cylindrically symmetric. The nonlinearity of the medium enters Maxwell’s equations by postulating a nonlinear material law $$D=\varepsilon E+\chi (x,y, \langle |E|^2\rangle )E$$ D = ε E + χ ( x , y , ⟨ | E | 2 ⟩ ) E between the electric field E, its time averaged intensity $$\langle |E|^2\rangle $$ ⟨ | E | 2 ⟩ and the electric displacement field D. We derive a new semilinear elliptic problem for the profiles $$U,{\widetilde{U}}:\mathbb {R}^2\rightarrow \mathbb {R}^3$$ U , U ~ : R 2 → R 3 $$\begin{aligned} Lu-V(x,y)u=f(x,y,u)\quad \hbox {with }u=\begin{pmatrix} U \\ {\widetilde{U}} \end{pmatrix}, \hbox { for }(x,y)\in \mathbb {R}^2, \end{aligned}$$ L u - V ( x , y ) u = f ( x , y , u ) with u = U U ~ , for ( x , y ) ∈ R 2 , where $$f(x,y,u)=\omega ^2\chi (x,y, |u|^2)u$$ f ( x , y , u ) = ω 2 χ ( x , y , | u | 2 ) u . Solving this equation we can obtain exact travelling wave solutions of the underlying nonlinear Maxwell equations. We are able to deal with super quadratic and subcritical focusing effects, e.g. in the Kerr-like materials with the nonlinear susceptibility of the form $$\chi (x,y,\langle |E^2|\rangle E) = \chi ^{(3)}(x,y)\langle |E|^2\rangle E$$ χ ( x , y , ⟨ | E 2 | ⟩ E ) = χ ( 3 ) ( x , y ) ⟨ | E | 2 ⟩ E . A variational approach is presented for the semilinear problem. The energy functional associated with the equation is strongly indefinite, since L contains an infinite dimensional kernel. The methods developed in this paper may be applicable to other strongly indefinite elliptic problems and other nonlinear phenomena.