Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bi-Hamiltonian systems on the dual of the Lie algebra of vector fields
  of the circle and periodic shallow water equations   This paper is a survey article on bi-Hamiltonian systems on the dual of the
Lie algebra of vector fields on the circle. We investigate the special case
where one of the structures is the canonical Lie-Poisson structure and the
second one is constant. These structures called affine or modified Lie-Poisson
structures are involved in the integrability of certain Euler equations that
arise as models of shallow water waves.

37
The categories of this paper are: supr-con
34
The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
33
The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
24
The categories of this paper are: dg-ga
21
The categories of this paper are: adap-org