A Remark of the Sanders-Wang's Theorem on Symmetry-integrability
Lizhou Chen
TL;DR
This work extends the symmetry-integrability analysis for scalar evolution equations to nonhomogeneous formal power series, showing that a single nontrivial symmetry yields an infinite hierarchy with orders constrained to four modular families modulo 6. It leverages order-estimation lemmas, the Gel'fand-Dikiĭ transform, and Beukers' theorem to construct higher-order symmetries via a pull-back mechanism, clarifying the algebraic structure of the symmetry space. Additionally, it proves polynomiality of generalized symmetries when the nonlinear part has degree below $m-1$, unifying known integrable cases like Burgers and KdV within a broader framework. The results delineate the possible symmetry families and their arithmetic progressions, contributing a rigorous classification to the theory of integrable evolution equations.
Abstract
We extend the integrability analysis for scalar evolution equations of type $$u_t=u_m+f(u,u_1,...,u_{m-1})$$ from the case that the right hand side is a $\lambda$-homogeneous formal power series to the case that it is a nonhomogeneous formal power series. It is proved that the existence of one nontrivial symmetry implies the existence of infinitely many, more precisely, the orders of the infinite integrable hierarchy must be one of the following cases: $\mathbb{Z}_++1$, $2\mathbb{Z}_++1$, $6\mathbb{Z}_+\pm1$, or $6\mathbb{Z}_++1$. Moreover, if the nonlinear part of the equation is a polynomial of order less than $m-1$, we show that any generalized symmetry is also of polynomial type.