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**Additional info for 1+1 Dimensional Integrable Systems**

**Example text**

Hence S is not diagonalizable. 145). 137). 137) are generalized to rational functions of λ, similar conclusions hold [121]. 4 KdV hierarchy, MKdV-SG hierarchy, NLS hierarchy and AKNS system with u(N ) reduction In the last section, we discussed the Darboux transformations for the AKNS system and more general systems. In those cases, we supposed that there were no reductions. In particular, there were no restrictions among the oﬀ-diagonal entries of P . However, in many cases, there are constraints on P and the Darboux transformation should keep those constraints.

283) are called the scattering data corresponding to (p, q), denoted by Σ(p, q). 284). 282). Property 6. N and N satisfy the follow system of linear integral equations (Gelfand-Levitan-Marchenko equations) ⎛ N (x, s) + B(2x + s) ⎝ ⎛ N (x, s) + B(2x + s) ⎝ ⎞ 1 0 0 1 +∞ ⎠ + ⎞ N (x, σ)B(2x + s + σ) dσ = 0, 0 +∞ ⎠ + N (x, σ)B(2x + s + σ) dσ = 0. 279) gives (p, q). The process to get scattering data from (p, q) is called the scattering process. It needs to solve the spectral problem of ordinary diﬀerential equations.

All the original eigenvalues are not changed, and ζ0 is a unique additional eigenvalue. 302) α (ζζ0 ) = 1/µ, hence ζ − ζ¯0 b(ζ) (ζ ∈ R), H ζ − ζ0 ζk − ζ¯0 Ck (k = 1, · · · , d), H Ck = ζk − ζ0 ζ0 − ζ¯0 . 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.