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


Mizrahi E. , Bilge A. H.

STUDIES IN APPLIED MATHEMATICS, vol.123, no.3, pp.233-255, 2009 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 123 Issue: 3
  • Publication Date: 2009
  • Doi Number: 10.1111/j.1467-9590.2009.00451.x
  • Title of Journal : STUDIES IN APPLIED MATHEMATICS
  • Page Numbers: pp.233-255

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 (1), (2), and (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) withk >= 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.