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Kac-Moody Lie algebras and soliton equations. II: Lax equations associated with \(A_ 1^{(1)}\). (English) Zbl 0643.35098

The paper is concerned with the role of Lie algebras in soliton theory. The authors concentrate on the soliton equations associated with the AKNS eigenvalue problem \[ V_ x(x,\zeta)\quad = \quad \left[\begin{pmatrix} -i&0\\ 0&i\end{pmatrix} \zeta + \begin{pmatrix} 0&q(x)\\ r(x)&0 \end{pmatrix} \right]V\quad (x,\zeta)=^{def}PV \] and on their relationship with the loop algebra of formal series \(\Sigma X_ k\zeta^{-k}\), \(X_ k\in sl(2).\)
Let \(V_ t=Q^{(N)}V\) be a time evolution for V, where \(Q^{(N)}=Q_ 0\zeta\) \(N+Q_ 1\zeta^{N-1}+...+Q_ N\) is a polynomial in \(\zeta\), \(Q_ s\in sl(2)\), \(Q=\sum^{\infty}_{j=0}Q_ j\zeta^{- j}=\sum^{\infty}_{j=0}\begin{pmatrix} h_ j & e_ j \\ f_ j & -h_ j\end{pmatrix} \zeta^{-j}\) is a formal series. All the \(Q_ j\) can be determined from the equation \(Q_ x=[P,Q].\)
Main result 1: Equations \(Q_{t_ k}=[Q^{(k)},Q]\), \(Q^{(k)}=\sum^{k}_{j=0}Q_ j\zeta^{k-j}\) are commuting Hamiltonian flows on the Lie algebra \(\{\sum^{\infty}_{0}X_ j\zeta^{-j}:\) \(X_ j\in sl(2,{\mathbb{C}})\}\), with respect to a natural Lie-Poisson bracket.
The next theorem shows that soliton equations in the generalized sense still have conservation laws.
Main result 2: There exist polynomials \(F_{k_ j}\) in the \(e_ i\), \(f_ i\), \(h_ i\), such that \(\partial F_{k_ j}/\partial t_ e=\ni F_{e_ k}/\partial t_ m.\)
[For part III see the following review.]
Reviewer: I.Chekalov

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Citations:

Zbl 0643.35099
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References:

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