Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algebras of commuting differential operators for integral kernels of
  Airy type   Differential operators commuting with integral operators were discovered in
the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic
expansions of the Fredholm determinants of integral operators arising in random
matrix theory. Very recently, it has been proved that all rational, symmetric
Darboux transformations of the Bessel, Airy, and exponential bispectral
functions give rise to commuting integral and differential operators [6, 7, 8],
vastly generalizing the known examples in the literature. In this paper, we
give a classification of the the rational symmetric Darboux transformations of
the Airy function in terms of the fixed point submanifold of a differential
Galois group acting on the Lagrangian locus of the (infinite dimensional) Airy
Adelic Grassmannian and initiate the study of the full algebra of differential
operators commuting with each of the integral operators in question. We
leverage the general theory of [8] to obtain explicit formulas for the two
differential operators of lowest orders that commute with each of the level one
and two integral operators obtained in the Darboux process. Moreover, we prove
that each pair of differential operators commute with each other. The commuting
operators in the level one case are shown to satisfy an algebraic relation
defining an elliptic curve.

24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
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The categories of this paper are: adap-org
18
The categories of this paper are: econ
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The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: eess
26
The categories of this paper are: bayes-an
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
1
The categories of this paper are: cs
15
The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an