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

We prove that arbitrary (nonpolynomial) scalar evolution equations of order $m\ge 7$, that are integrable in the sense of admitting the canonical conserved densities $\ro^{(1)}$, $\ro^{(2)}$, and $\ro^{(3)}$ introduced in [MSS,1991], 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 $k\ge m-2$ 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.

💡 Research Summary

The paper addresses the long‑standing problem of classifying scalar evolution equations of high order that are not assumed to be polynomial or scale‑invariant. Using the formal symmetry approach pioneered by Mikhailov, Shabat and Sokolov, the authors focus on the three canonical conserved densities ρ^(1), ρ^(2) and ρ^(3) introduced in the 1991 monograph. For an evolution equation u_t = F