Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Travelling waves for Maxwell's equations in nonlinear and nonsymmetric
  media   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 is
not necessarily cylindrically symmetric. The nonlinearity of the medium enters
Maxwell's equations by postulating a nonlinear material law $D=\varepsilon
E+\chi(x,y, \langle |E|^2\rangle)E$ between the electric field $E$, its time
averaged intensity $\langle |E|^2\rangle$ and the electric displacement field
$D$. We derive a new semilinear elliptic problem for the profiles $U,\widetilde
U:\mathbb{R}^2\to\mathbb{R}^3$ $$Lu-V(x,y)u=f(x,y,u)\quad\hbox{with
}u=\begin{pmatrix} U \\ \widetilde U \end{pmatrix}, \hbox{ for
}(x,y)\in\mathbb{R}^2,$$ where $f(x,y,u)=\omega^2\chi(x,y, |u|^2)u$. 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 $\chi(x,y,\langle |E^2|\rangle E) =
\chi^{(3)}(x,y)\langle |E|^2\rangle E$. 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.

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The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: cs
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org