'level Grading' a New Graded Algebra Structure on Differential Polynomials: Application To the Classification of Scalar Evolution Equations

Abstract

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.