Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Electromagnetic Shocks in Quantum Vacuum   The interaction of two counter-propagating electromagnetic waves in a vacuum
is analyzed within the framework of the Heisenberg-Euler formalism in quantum
electrodynamics. The nonlinear electromagnetic wave in the quantum vacuum is
characterized by wave steepening, subsequent generation of high order harmonics
and electromagnetic shock wave formation with electron{positron pair generation
at the shock wave front.

30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
2
The categories of this paper are: math
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn
4
The categories of this paper are: cond-mat