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**Example text**

Although it is special, it is very useful. 5 Darboux transformation and scattering, inverse scattering theory The scattering and inverse scattering theory is an important part of the soliton theory. It transforms the problem of solving the Cauchy problem of a nonlinear partial diﬀerential equation to the problem of describing the spectrum and eigenfunctions of the Lax pair. Here we consider the 2 × 2 AKNS system as an example to show the outline of the scattering and inverse scattering theory (see [23] for details).

161) The ﬁrst two equations imply 1 A = − Bx , 2 C = ζB − uB − 12 Bxx . 161) we get 1 ut = −2(ζ − u)Bx + ux B + Bxxx . 165) 36 DARBOUX TRANSFORMATIONS IN INTEGRABLE SYSTEMS 1 ut = 2ubn,x + ux bn + bn,xxx . 166) is the equation of u. When n ≥ 2, it is called a KdV equation of higher order. 165) and b00 [0] = 1, b0j [0] = 0 (j ≥ 1). 165) uniquely. The ﬁrst few b0j ’s are 1 b01 = u, 2 1 3 b02 = uxx + u2 , · · · . 168) The corresponding equations are n = 0: Linear equation ut = α0 (t)ux . 169) n = 1: 1 3 uxxx + uux + α1 (t)ux .

234) ¯0. S − S ∗ = λ0 − λ Therefore, under the action of the Darboux matrix λI − S, U is transformed to ⎛ U = (λI − S)U (λI − S)−1 − Sx (λI − S)−1 = ⎝ where p = p + 2S12 = p + ⎞ λ p −¯ p −λ ¯ 0 )¯ 2(λ0 + λ σ . 11), we know that V = (λI − S)V (λI − S)−1 − St (λI − S)−1 is also a polynomial ∗ ¯ = −V (λ) holds. Moreover, λI − S gives a Darboux of λ and V (−λ) transformation from an equation in the nonlinear Schr¨ o¨dinger hierarchy to the same equation. This leads to the following theorem. 226) for λ = λ0 .