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Toward the Classification of Scalar Nonpolynomial Evolution Equations: Polynomiality in Top Three Derivatives
Author(s) -
Mizrahi Eti,
Bilge Ayşe Hümeyra
Publication year - 2009
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2009.00451.x
Subject(s) - mathematics , integrable system , scalar (mathematics) , homogeneous , pure mathematics , polynomial , evolution equation , mathematical physics , mathematical analysis , algebra over a field , combinatorics , geometry
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 .