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

Jarosław Mederski, Jacopo Schino

Abstract

We look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+ωt)+ \widetilde U(x,y)\sin(kz+ωt),\quad (x,y,z)\in\mathbb{R}^3,\, t\in\mathbb{R}, $$ 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.

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

Abstract

We look for travelling wave fields 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.
Paper Structure (3 sections, 13 theorems, 118 equations)

This paper contains 3 sections, 13 theorems, 118 equations.

Table of Contents

  1. Variational setting
  2. Analysis of Cerami sequences
  3. Non-TE modes

Key Result

Theorem 1.1

Suppose that (V), (F0)--(F3) are satisfied and $F$ is radial. Then there exist infinitely many solutions to eq of the form $u_n=v_n+w_n$ such that $J(u_n)\to\infty$ as $n\to\infty$, $v_n\in H^1(\mathbb{R}^2)^6$, $v_n\neq 0$, $w_n\in L^2(\mathbb{R}^2)^6$, $\Phi(w_n) \in L^1(\mathbb{R}^2)$, $Lw_n=0$, for some radial functions $\alpha_n,\widetilde{\alpha}_n,\gamma_n,\widetilde{\gamma}_n:\mathbb{R}^2

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 14 more