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

2009/09/25 by Eti Mizrahi, Ayşe Hümeyra Bilge
Physics and Astronomy · Mathematics · #Nonlinear Waves and Solitons #Advanced Differential Equations and Dynamical Systems #Fractional Differential Equations Solutions

paper · doi:10.1111/j.1467-9590.2009.00451.x

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 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 .

Related

• Toward the Classification of Scalar Nonpolynomial Evolution Equations:Polynomiality in Top Three Derivatives
• On the Classification of Scalar Non-Polynomial Evolution Equations: Quasilinearity
• A New Graded Algebra Structure on Differential Polynomials: Level Grading and its Application to the Classification of Scalar Evolution Equations in 1+1 Dimension
• A Remark of the Sanders-Wang's Theorem on Symmetry-integrability
• On the Classification of Fifth Order Quasi-linear Non-constant Separant Scalar Evolution Equations of the KdV-type