A New Graded Algebra Structure on Differential Polynomials: Level Grading and its Application to the Classification of Scalar Evolution Equations in 1+1 Dimension

TL;DR

This paper introduces a new level grading on the algebra of differential polynomials, enabling a systematic classification of scalar evolution equations in 1+1 dimensions, especially those of KdV-type with conserved densities.

Contribution

The paper defines a novel level grading that preserves algebraic structures under differentiation and integration, facilitating the classification of integrable evolution equations.

Findings

Established the properties of the level grading on differential polynomials.

Applied the grading to classify scalar evolution equations of KdV-type.

Determined the top level parts of conserved densities for integrable equations.

Abstract

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 $m=5,7,9,11,13,15$.