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C+ = {z ∈ C | Im ζ > 0}, C− = {z ∈ C | Im ζ < 0}. 52 DARBOUX TRANSFORMATIONS IN INTEGRABLE SYSTEMS Property 1. 265) ( Im ζ ≤ 0). Moreover, ψr , ψl (resp. ψr , ψl ) are continuous for ζ ∈ C+ ∪ R (resp. z ∈ C− ∪ R), and holomorphic with respect to ζ in C+ (resp. C− ). These solutions are called Jost solutions. If ζ ∈ R, then ψl and ψl are linearly independent. 266) ψ r = r+ ψ l + r− ψ l . Considering the Wronskian determinant between ψr , ψl and the Wronskian determinant between ψr , ψl , we have Property 2.

253), we obtain ¯ + S)∗ (λI − S) = (λ − µ)(λ + µ ¯)I. 255) m Vj λ m−j −1 (λI − S) −1 + (λI − S)t (λI − S) , j=0 m ( Vj λm−j )∗ j=0 = (λI − S)∗−1 =− ¯ m−j (λI − S)∗ + (λI − S)∗−1 (λI − S)∗ Vj∗ λ t j=0 m ¯ m−j (λI ¯ + S)−1 − (λI ¯ + S)t (λI ¯ + S)−1 Vj (−λ) ¯ + S) = −(λI m m j=0 ¯ m−j . 256) ¯ = −V (λ)∗ . Likewise, U (−λ) ¯ = −U (λ)∗ . The theorem Hence V (−λ) is proved. 3, a Darboux transformation of higher degree can be derived by the composition of Darboux transformations of degree one. However, with the u(N ) reduction, we have also the following special and more direct construction [117, 17].

72) for P . 139). 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 .