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

TL;DR

This paper investigates traveling wave solutions to Maxwell's equations in nonlinear, cylindrically symmetric media, presenting new solutions with diverging energy and exploring more general nonlinearities using N-functions.

Contribution

It introduces a new sequence of solutions with diverging energy and extends the analysis to more general nonlinearities via N-functions.

Findings

New solutions with diverging energy are constructed.

The solutions differ from previous work by McLeod, Stuart, and Troy.

General nonlinearities are considered through N-functions.

Abstract

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 and cylindrically symmetric medium. We obtain a sequence of solutions with diverging energy that is different from that obtained by McLeod, Stuart, and Troy. In addition, we consider a more general nonlinearity, controlled by an \textit{N}-function.