‘Level grading’ a new graded algebra structure on differential polynomials: application to the classification of scalar evolution 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 = ∂ k+i u/∂x k+i over the ring K (k) of C ∞ 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 ρ is level homogeneous over K (k) , then the top level part of D t ρ 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.

On the classification of scalar evolution equations with non-constant separant

“…Then, in [8] we showed that evolution equations with non-trivial ρ (i) , i = 1, 2, 3 are polynomial in u m−1 and u m−2 and possess a certain scaling property that we called "level grading" [9]. For m = 5, we have shown that there is a candidate for non-quasilinear integrable equation [1], we obtained canonical densities ρ (i) , i = 1, .…”

“…The parts that has the highest level is called the "Top Level" part of P and the dependency of the coefficients of the top level on u b is called their "Top Dependency" [9].…”

“…We aimed to obtain a similar uniqueness result for general, non-polynomial integrable equations and we applied the method of formal symmetries, based on the existence of "canonical densities" [7], to the classification of integrable evolution equations in 1 + 1 dimensions, u t = F [u] [1,8,11,9].…”