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We look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+\omega t)+\widetilde U(x,y)\sin(kz+\omega t),\quad (x,y,z)\in\mathbb{R}^3,\,t\in\mathbb{R} $$ satisfying Maxwell's equations in a nonlinear medium which isnot necessarily cylindrically symmetric. The nonlinearity of the medium entersMaxwell's equations by postulating a nonlinear material law $D=\varepsilonE+\chi(x,y, \langle |E|^2\rangle)E$ between the electric field $E$, its timeaveraged intensity $\langle |E|^2\rangle$ and the electric displacement field$D$. We derive a new semilinear elliptic problem for the profiles $U,\widetildeU:\mathbb{R}^2\to\mathbb{R}^3$ $$Lu-V(x,y)u=f(x,y,u)\quad\hbox{with}u=\begin{pmatrix} U \\ \widetilde U \end{pmatrix}, \hbox{ for}(x,y)\in\mathbb{R}^2,$$ where $f(x,y,u)=\omega^2\chi(x,y, |u|^2)u$. Solvingthis equation we can obtain exact travelling wave solutions of the underlyingnonlinear Maxwell equations. We are able to deal with super quadratic andsubcritical focusing effects, e.g. in the Kerr-like materials with thenonlinear susceptibility of the form $\chi(x,y,\langle |E^2|\rangle E) =\chi^{(3)}(x,y)\langle |E|^2\rangle E$. A variational approach is presented forthe semilinear problem. The energy functional associated with the equation isstrongly indefinite, since $L$ contains an infinite dimensional kernel. Themethods developed in this paper may be applicable to other strongly indefiniteelliptic problems and other nonlinear phenomena.

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