2012/04/14 by Eti Mizrahi, E. Mizrahi, Ayşe Hümeyra Bilge +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Waves and Solitons #Numerical methods for differential equations #nlin.SI
paper · pdf · doi:10.48550/arxiv.1204.3171
arxiv created 2012/04/14 · openalex publication_date 2012/04/14 · arxiv updated 2012/04/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We define a new grading, that we call the "level grading", on the algebra of polynomials generated by the derivatives uk+i=∂k+iu/∂ xk+i over the ring K(k) of C∞ functions of u,u1,...,uk. This grading has the property that the total derivative and the integration by parts with respect to x are filtered algebra maps. In addition, if u satisfies an evolution equation ut=F[u] and F is a level homogeneous differential polynomial, then the total derivative with respect to t, Dt, is also a filtered algebra map. Furthermore if ρ is level homogeneous over K(k), then the top level part of Dtρ depends on uk only. This property allows to determine the dependency of F[u] on uk from the top level part of the conserved density conditions. We apply this structure to the classification of "level homogeneous" scalar evolution equations and we obtain the top level parts of integrable evolution equations of "KdV-type", admitting an unbroken sequence of conserved densities at orders m=5,7,9,11,13,15.