A Remark of the Sanders-Wang's Theorem on Symmetry-integrability

TL;DR

This paper extends symmetry-integrability analysis for scalar evolution equations to nonhomogeneous cases, establishing conditions for infinite symmetries and polynomiality of generalized symmetries.

Contribution

It generalizes Sanders-Wang's theorem to nonhomogeneous power series and classifies the possible orders of integrable hierarchies.

Findings

Existence of one nontrivial symmetry implies infinitely many.

Orders of integrable hierarchies are restricted to specific sets.

Generalized symmetries are polynomial if the nonlinear part is polynomial.

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.