标量非多项式发展方程的分类：前三阶导数的多项式性
Toward the Classification of Scalar Nonpolynomial Evolution Equations:Polynomiality in Top Three Derivatives
摘要：
我们证明任意（非多项式）$m\ge 7$阶标量演化方程，在承认[MSS，1991]中引入的正则守恒密度$\ro^{(1)}$、$\ro^{(2)}$和$\ro^{(3)}$的意义上是可积的，我们还在$u_{m-i}$中的函数环上引入了$k\ge m-2$中多项式代数的分次，并证明了可积方程相对于该分级是尺度齐次的。
英文摘要：
We prove that arbitrary (nonpolynomial) scalar evolution equations of order $m\ge 7$, that are integrable in the sense of admitting the canonical conserved densities $\ro^{(1)}$, $\ro^{(2)}$, and $\ro^{(3)}$ introduced in [MSS,1991], 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 $k\ge m-2$ 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.
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