Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A New Graded Algebra Structure on Differential Polynomials: Level
  Grading and its Application to the Classification of Scalar Evolution
  Equations in 1+1 Dimension   We define a new grading, that we call the "level grading", on the algebra of
polynomials generated by the derivatives $u_{k+i}=\partial^{k+i}u/\partial
x^{k+i}$ over the ring $K^{(k)}$ of $C^{\infty}$ functions of $u,u_1,...,u_k$.
This grading has the property that the total derivative and the integration by
parts with respect to $x$ are filtered algebra maps. In addition, if $u$
satisfies an evolution equation $u_t=F[u]$ and $F$ is a level homogeneous
differential polynomial, then the total derivative with respect to $t$, $D_t$,
is also a filtered algebra map. Furthermore if $\rho$ is level homogeneous over
$K^{(k)}$, then the top level part of $D_t\rho$ depends on $u_k$ only. This
property allows to determine the dependency of $F[u]$ on $u_k$ from the top
level part of the conserved density conditions. We apply this structure to the
classification of "level homogeneous" scalar evolution equations and we obtain
the top level parts of integrable evolution equations of "KdV-type", admitting
an unbroken sequence of conserved densities at orders $m=5,7,9,11,13,15$.

20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
35
The categories of this paper are: mtrl-th
10
The categories of this paper are: q-bio