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

摘要

We define a new grading, that we call the "level grading", on the algebra ofpolynomials generated by the derivatives $u_{k+i}=\partial^{k+i}u/\partialx^{k+i}$ 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 byparts 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 homogeneousdifferential 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. Thisproperty allows to determine the dependency of $F[u]$ on $u_k$ from the toplevel part of the conserved density conditions. We apply this structure to theclassification of "level homogeneous" scalar evolution equations and we obtainthe top level parts of integrable evolution equations of "KdV-type", admittingan unbroken sequence of conserved densities at orders $m=5,7,9,11,13,15$.