'Level grading' a new graded algebra structure on differential polynomials: application to the classification of scalar evolution equations

We define a new grading, which we call the 'level grading', on the algebra of polynomials generated by the derivatives u(k+i) over the ring K-(k) of C-infinity functions of x, t, u, u(1), ..., u(k), where . 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 the evolution equation u(j) = F[u], where F is a polynomial of order m = k + p and of level p, then the total derivative with respect to t, D-t, is also a filtered algebra map. Furthermore, if the separant partial derivative F/partial derivative u(m) belongs to K-(k), then the canonical densities (i) are polynomials of level 2i + 1 and (i) is of level 2i + 1 + m. We define 'KdV-like' evolution equations as those equations for which all the odd canonical densities rho((i)) are non-trivial. We use the properties of level grading to obtain a preliminary classification of scalar evolution equations of orders m = 7, 9, 11, 13 up to their dependence on x, t, u, u(1) and u(2). These equations have the property that the canonical density rho((-1)) is (alpha u(3)(2) + beta u(3) + gamma)(1/2) where alpha, beta and gamma are functions of x, t, u, u(1), u(2). This form of rho((-1)) is shared by the essentially nonlinear class of third order equations and a new class of fifth order equations.