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By Casey J.

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147) is ⎛ (λI − HΛH −1 ) ⎝ eλx−2iλ Take 2t e−λx+2iλ ⎛ ⎛ H( ) = ⎝ 0 0 Λ( ) = ⎝ then we can choose ⎞ 2t ⎠. 154) 2 ( ) h22 = − sech(αx)e iα t − tanh(αx), θ = x − 2i 2 t. 155) ⎞ sinh(αx)e−iα t e−2iα t α ⎠, S= ⎝ 2 ∆ − sinh2 (αx) − sinh(αx)e−iα t 2 p = where 2αe−2iα t , ∆ q = 2α sinh2 (αx) ∆ 2 ∆ = (α + 2 − 2 sech(αx)e−iα t ) cosh2 (αx). 158) Note that both eigenvalues of S are zero, but S = 0. Hence S is not diagonalizable. 145). 137). 137) are generalized to rational functions of λ, similar conclusions hold [121].

131), S2 − S1 is non-degenerate. The Darboux matrix of degree two is D(λ) = (λI − S2 )(λI − S1 ) = λ2 I − λ(S22 − S12 )(S2 − S1 )−1 + (S2 − S1 )S2 (S2 − S1 )−1 S1 . 136) It is easy to check that D(λ) is symmetric to S1 and S2 . Therefore, we can also obtain the theorem of permutability by this symmetry. 9 can be applied not only to the AKNS system, but also to many other evolution equations, especially to the Lax pairs whose U and V are polynomials of λ. On the other hand, we also show that those Darboux matrices include all the diagonalizable Darboux matrices of the form λI − S, and any non-diagonalizable Darboux matrix can be obtained from the limit of diagonalizable Darboux matrices.

Let Sj = Hj Λj Hj−1 and denote (α) (α) Λα = diag(λ1 , · · · , λN ), (α) (α) Hα = (h1 , · · · , hN ). (2) (2) (2) After the action of λI −S1 , hj is transformed to (λj I −S1 )hj . Hence H2 . The second H2 is transformed to H2 = H2 Λ2 − S1 H2 = (S2 − S1 )H Darboux matrix of degree one is λI − S2 where S2 = H2 Λ2 H2−1 = (S2 − S1 )S2 (S2 − S1 )−1 . 131), S2 − S1 is non-degenerate. The Darboux matrix of degree two is D(λ) = (λI − S2 )(λI − S1 ) = λ2 I − λ(S22 − S12 )(S2 − S1 )−1 + (S2 − S1 )S2 (S2 − S1 )−1 S1 .