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

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

2009-09-08

arxiv.org/abs/0909.1519v1

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Accepted to publish in "Studies in Applied Mathematics"

Scientific paper

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.

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