Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dark equations   Observing the Universe, astronomers have concluded that the motion of stars
can not be accounted for unless one assumes that most of the mass in the
Universe is carried on by a ``dark matter", so far impervious to all attempts
at being detected. There is now a similar concept of ``dark energy". I shall
discuss a different subject, ``dark equations". These have never indicated that
they influence anything or even exist, but if one supposes that they do exist,
one can systematically discover them and study their properties, some of which
turn out to be strange and others mysterious. These equations are similar in
spirit to what one gets when linearizing a given system, or studies how an
external linear wave interacts with a particular solution of a given system. We
define and study linear extensions of dynamical systems in general, and
integrable and Hamiltonian systems in particular. Systems discussed include the
KdV and mKdV equations and the associated Miura maps, the Burgers hierarchy and
the associated Hopf--Cole transformations, long wave equations, the Benney
hierarchy, and the KP hierarchy.

8
The categories of this paper are: hep-ex
32
The categories of this paper are: atom-ph
22
The categories of this paper are: chao-dyn
35
The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
29
The categories of this paper are: funct-an
21
The categories of this paper are: adap-org
2
The categories of this paper are: math
1
The categories of this paper are: cs
17
The categories of this paper are: q-fin