Sanders-Wang对称可积定理的一个注记
A Remark of the Sanders-Wang's Theorem on Symmetry-integrability
摘要：
将$$u_t=u_m+f(u,u_1,...,u_{m-1})$$型标量发展方程的可积性分析从右端为$λ$ -齐次形式幂级数的情形推广到右端为非齐次形式幂级数的情形.证明了一个非平凡对称的存在意味着无穷多个对称的存在，更确切地说，无穷可积族的阶必须是下列情形之一：$\mathbb{Z}_++1$，$2\mathbb{Z}_++1$，$6\mathbb{Z}_+\pm1$，或$6\mathbb{Z}_++1$ .此外，如果方程的非线性部分是阶数小于$m-1$的多项式，我们证明了任何广义对称也是多项式型的。
英文摘要：
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 $λ$-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.
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