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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 . 72) for P . 139). 9 also holds. 137) for λ = λi (i = 1, 2, · · · , N ) such that H = (h1 , · · · , hN ) and S = HΛH −1 .

18 n = 1, m = 2, α0 = −4, α1 = 0, β0 = 0, β1 = 1, then the equation becomes the equation describing one-dimensional nonlinear lattice of atoms [70] 3 uxt + u2x uxx + uxxxx − sin u = 0. 217) ⎛ ⎞ α Now we consider the Darboux transformation. 192) for λ = −λ0 . 192) for λ = λ0 . 219) ⎞ cos θ sin θ sin θ − cos θ ⎠. 220) From σx = −p(1 + σ 2 ) − 2λ0 σ, we have θx = −2p − 2λ0 sin θ. 224) or equivalently, for suitable choice of the integral constant. It remains to prove that the Darboux matrix λI − S keeps the reduction of MKdV-SG hierarchy.

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. For ζ ∈ R, r− (ζ) = R1 (x, ζ)L2 (x, ζ) − R2 (x, ζ)L1 (x, ζ), r+ (ζ) = R2 (x, ζ)L1 (x, ζ) − R1 (x, ζ)L2 (x, ζ). 267) r− (ζ) can be holomorphically extended to C+ ∪ R, and r+ (ζ) can be holomorphically extended to C− ∪ R. , R = (R1 , R2 )T . L1 , L2 , R1 , R2 , L1 , L2 have the similar meanings. 53 1+1 dimensional integrable systems The asymptotic properties of the four Jost solutions in Property 1 as x → ±∞ are listed in the next property.

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