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282) b(ζ)e−iζy dζ. 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.

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 . 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.

For the KdV equation, the problem can be solved similarly, but the scattering and inverse scattering theory is simpler. 1 Outline of the scattering and inverse scattering theory for the 2 × 2 AKNS system First, we give the definition of the scattering data for the 2×2 complex AKNS system. 48) becomes ⎛ Φx = ⎝ ⎞ −iζ p q iζ ⎠ Φ. 261) Suppose p, q and their derivatives with respect to x decay fast enough at infinity. Let C be the complex plane and R be the real line. , C+ = {z ∈ C | Im ζ > 0}, C− = {z ∈ C | Im ζ < 0}.

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1+1 Dimensional Integrable Systems


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