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142) and they satisfy Ut − Vx + [U , V ] = 0. 141) holds. If S is obtained, we have the Darboux transformation (U, V, Φ) → (U , V , Φ ). 8 . 137) if and only if S satisfies Sx + [S, U (S)] = 0, St + [S, V (S)] = 0. 145) Here m U (S) = j=0 Uj S m−j , n V (S) = Vj S n−j . 139). 137) for λ = λi , H = (h1 , · · · , hN ). If det H = 0, let S = HΛH −1 , then the following theorems holds. 9 . 137). 10 . 145) is integrable. The proofs are omitted since they are similar to the proofs for the corresponding theorems above.

9 also holds. 137) for λ = λi (i = 1, 2, · · · , N ) such that H = (h1 , · · · , hN ) and S = HΛH −1 . 137) but it can not be diagonalized at any points, then there exist a series of Darboux matrices λI − Sk such that Sk ’s and their derivatives with respect to x and t converge to S and its derivatives respectively. 10. 15 An example of a Darboux matrix which is not diagonalizable everywhere. 147) ⎠Φ 2iλ2 − ipq i whose integrability condition leads to the nonlinear evolution equations i t = pxx − 2p2 q, ip −iqt = qxx − 2pq 2 .

Moreover, λI − S gives a Darboux of λ and V (−λ) transformation from an equation in the nonlinear Schr¨ o¨dinger hierarchy to the same equation. This leads to the following theorem. 226) for λ = λ0 . 226). 238) of the same equation. 227) imply ¯ = −U (λ)∗ , U (−λ) ¯ = −V (λ)∗ . 239) Here we generalize it to the AKNS system. 226) has u(N ) reduction, because U (λ) and V (λ) are in the Lie algebra u(N ) when λ is purely imaginary. This is a very popular reduction. We want to construct Darboux matrix which keeps u(N ) reduction.

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1+1 Dimensional Integrable Systems

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