By Gu C., Hu H., Zhou Z.
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We consider its spectrum in L2 (R) × L2 (R). 267) implies that ψr and ψl are linearly dependent. Hence ψr → 0 as x → ±∞. Similarly, if ζ ∈ C− and r+ (ζ) = 0, then ψr and ψl are linearly dependent. Hence ψr → 0 as x → ±∞. Since r− (ζ) and r+ (ζ) are holomorphic in C+ and C− respectively, their zeros are discrete. These zeros are the eigenvalues of L. The set of all eigenvalues of L is denoted by IP σ(L). 270) has a nontrivial bounded solution. σ(L) = R ∪ IP σ(L) is called the spectrum of the operator L.
177) is a Darboux transformation from any equation in the KdV hierarchy to the same equation. 1. 159) is ⎛ R−1 (λ0 ) ⎝ ⎞ α β ⎛ ⎠=⎝ −β α + λ0 β ⎞ ⎠. 182) ⎞ ⎠. 159), let ⎛ D = R−1 ⎝ ⎛ =⎝ ⎞ 1 0 0 −1 ⎛ ⎠ (λI − S)R = ⎝ ⎞ −σ 1 ζ − ζ0 + σ 2 −σ ⎠ −σ ⎞ 1 λ2 − λ20 + σ 2 −σ ⎠ (ζζ0 = λ20 ). 184) Then U = DU D−1 + Dx D−1 ⎛ =⎝ ⎞ 0 1 ζ − 2ζζ0 + u + 2σ 2 0 ⎠. 179)). , the Darboux transformation keeps the t part invariant. 159). 188) u = 2ζζ0 − u − 2σ 2 of the same equation. 184) is chosen as D = R−1 ⎝ 0 −1 not R−1 (λI − S)R.
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. Note that for the AKNS system, we can solve Vi [P ]’s from a system of diﬀerential equations by choosing “integral constants” and these Vi [P ]’s are diﬀerential polynomials of P .
Darboux Transformations in Integrable Systems. Theory and their Applications to Geometry by Gu C., Hu H., Zhou Z.