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

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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  um− i  for i = 0, 1, 2. We also introduce a grading in the algebra of polynomials in  uk   with  k ≥ m − 2   over the ring of functions in  x, t, u, … , um−3   and show that integrable equations are scale homogeneous with respect to this grading.
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Document Type: Research Article

Affiliations: Istanbul Technical University

Publication date: October 1, 2009

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