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An explicit expression for the Korteweg-de Vries hierarchy. (English) Zbl 0659.35089

The prototypical Korteweg-de Vries (KdV) hierarchy of equations reads \(u_ t=dG_ n(u),\) \(n=1,2,...\), where \(D=\partial /\partial x\), \(G_ 1=u\) and \(G_{n+1}=(D^ 2+4u-2D^{-1}u_ x)G_ n.\) To solve this recursion equation for an explicit expression of \(G_ n\) is far from easy. Based on the Gel’fand and Dikii’s work, the author obtains the following nice explicit formula for \(G_ n:\) \[ n!((2n-1)!)^{-1} G_ n(u)=((q_ 2+n-1)(q_ 3+n-2)...(q_ n+1))^{-1}\cdot \]
\[ \cdot C(0,q_ 2)C(q_ 2,q_ 3)...C(q_{n-1},q_ n)u_{-q_ 2}u_{q_ 2-q_ 3}...u_{q_{n-1}-q_ n}u_{q_ n}, \] where \(u_ p=D^ pu\) with the convention that \(u_{-1}=0\), \(u_{-2}=1\) and \(C(p,q)=\left( \begin{matrix} p\\ q\end{matrix} \right)+\delta_ q^{p+2},\) and the sum is taken over all integers \(q_ 2,q_ 3,...,q_ n\) such that \(0\leq q_ 2\leq 2\), \(0\leq q_ 3\leq q_ 2+2,...,0\leq q_ n\leq q_{n-1}+2\). To the reviewer’s memory the explicit form of the KdV hierarchy was also obtained elsewhere in literature, however the above-mentioned form is indeed more elegant.
Reviewer: G.-Z.Tu

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
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