Toward the Classification of Scalar Nonpolynomial Evolution Equations: Polynomiality in Top Three Derivatives

Abstract

We prove that arbitrary (nonpolynomial) scalar evolution equations of order m ≥ 7, that are integrable in the sense of admitting the canonical conserved densities ρ (1) , ρ (2) , and ρ (3) introduced in [1], are polynomial in the derivatives u m−i for i = 0, 1, 2. We also introduce a grading in the algebra of polynomials in u k with k ≥ m − 2 over the ring of functions in x, t, u, . . . , u m−3 and show that integrable equations are scale homogeneous with respect to this grading.

On the Classification of Fifth Order Quasi-linear Non-constant Separant Scalar Evolution Equations of the KdV-Type

“…The notions of "level grading" and level homogeneity introduced in [6] are analogues of scaling and scale homogeneity for polynomials; in the level grading context, the "level above a base level k" reflects the number of differentiations applied to a function that depends on the derivatives of order at most k. The crucial property of the level grading is its invariance under integrations by parts. This property implies that "top level" terms of a conserved density ρ will give top level terms in its time derivative D t ρ, hence we may omit lower level terms and still track correctly higher level integrability conditions.…”

“…In a series of papers we have applied the "formal symmetry" method of [1] to generic, non-polynomial equations; we have first proved that integrable evolution equations of order 7 and greater are quasi-linear [5]; then we have shown that they are polynomial in top three derivatives [6]. Motivated by these polynomiality results, we decided to investigate the structure of fifth order quasi-linear integrable equations with non-constant separant to see whether they would be polynomial and transformable to the constant separant case classified in [1].…”

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

“…Although we have obtained polynomiality in top 3 derivatives in [8], we start here with the quasilinear form and indicate the steps towards the classification of lower order evolution equations.…”

“…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, .…”

“…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].…”