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Additional info for 1+1 Dimensional Integrable Systems

Example text

292) For a linear equation, if “scattering” is changed to “Fourier transformation” and “inverse scattering” is changed to “inverse Fourier transformation” in the above diagram, then it becomes the diagram for solving the initial value problem by Fourier transformations which has been used extensively for linear problems. Therefore, the scattering and inverse scattering method can be regarded as a kind of Fourier method for nonlinear problems. 2 Change of scattering data under Darboux transformations for su(2) AKNS system For the AKNS system, the scattering data include {ζk , Ck , ζk , Ck , b(ζ), b(ζ)}.

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. 166). 189) φt = Aφ + Bφx . 1. However, here A and B can be polynomials of ζ of arbitrary degrees, whose coeﬃcients are diﬀerential polynomials of u. 1 was a special case. 16, b = ax , hence σ = ax /a. 186) give the original Darboux transformation u = u + 2(ln a)xx . 165), we can get b0 , b1 , · · · recursively, whose integral constants can be functions of t.

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).