Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Kinetic theory and Lax equations for shock clustering and Burgers
  turbulence   We study shock statistics in the scalar conservation law $\partial_t u +
\partial_x f(u)=0$, $x \in \R$, $t>0$, with a convex flux $f$ and spatially
random initial data. We show that the Markov property (in $x$) is preserved for
a large class of random initial data (Markov processes with downward jumps and
derivatives of L\'evy processes with downward jumps). The kinetics of shock
clustering is then described completely by an evolution equation for the
generator of the Markov process $u(x,t)$, $x\in \R$. We present four distinct
derivations for this evolution equation, and show that it takes the form of a
Lax pair. The Lax equation admits a spectral parameter as in Manakov (1976),
and has remarkable exact solutions for Burgers equation ($f(u)=u^2/2$). This
suggests the kinetic equations of shock clustering are completely integrable.

1
The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
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The categories of this paper are: hep-th
2
The categories of this paper are: math
30
The categories of this paper are: q-alg
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The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
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The categories of this paper are: eess
12
The categories of this paper are: nucl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: comp-gas