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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. That is, after the action of the Darboux matrix, the derived potentials U (λ) and V (λ) must satisfy ¯ = −U (λ)∗ , U (−λ) ¯ = −V (λ)∗ . 94) can not be arbitrary. They should satisfy the following two conditions: µ where µ is a complex number (µ (1) λ1 , · · · , λN can only be µ or −¯ is not real).

303) (2) If ζ0 is an eigenvalue: ζ0 = ζj , and µ = α(ζζj ), then, after the action of the Darboux transformation, ζ0 is no longer an eigenvalue. 304) r+ (ζ) = r+ (ζ) (ζ ∈ R), α (ζk ) = α(ζk ) (k = 1, · · · , d, k = j), ζ − ζ0 b(ζ) (ζ ∈ R), H ζ − ζ¯0 ζk − ζ0 Ck = Ck (k = 1, · · · , d, k = j). 305) Proof. (1) ζ0 ∈ IP σ(L). 299) are not 0. Property 3 implies lim σ = ∞, lim σ = 0. 307) ⎠. Under the action of the Darboux transformation, the Jost solutions are changed to 1 (−iζI − S)ψr (x, t, ζ), −iζ + iζ¯0 1 (−iζI − S)ψl (x, t, ζ).

2) ζ0 = ζj ∈ IP σ(L), µ = α(ζζj ). 319) ⎞ −iζ + iζζ0 0 0 −iζ + iζ¯0 −iζ + iζ¯0 0 0 −iζ + iζζ0 ⎠, ⎞ ⎠. 320) 63 1+1 dimensional integrable systems Under the action of the Darboux transformation, the Jost solutions become 1 (−iζI − S)ψr (x, t, ζ), −iζ + iζζ0 1 ψl (x, t, ζ) = (−iζI − S)ψl (x, t, ζ). 322) ⎞ 0 ⎟ r− (ζ) ⎠. 324) (ζ ∈ R), and ζ − ζ0 b(ζ). 324) we know that the Darboux transformation removes the eigenvalue ζ0 (= ζj ). If ζ = ζk (k = j), then ψr = α(ζk )ψl , hence b (ζ) = α (ζk ) = α(ζk ), dr− (ζk ) ζk − ζ0 = Ck .

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