By Gu C., Hu H., Zhou Z.

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**Example text**

After the transformation of λI − S1 , P → P = P + [J, S1 ]. Then after the transformation of λI −S2 , P → P = P +[J, S2 ], and so on. Hence, after the transformation of D, P → P + [J, S1 + · · · + Sr ] = P − [J, D1 ]. 72). (4) is proved. This proves the lemma. Darboux transformation has an important property — the theorem of permutability. This theorem originated from the B¨ a¨cklund transformation of the sine-Gordon equation and there are a lot of generalizations and various proofs. The proof here is given by [52] (2 × 2 case) and [33] (N × N case).

Since any Darboux transformation without explicit expression is a limit of Darboux transformations with explicit expressions, the theorem of permutability also holds for the Darboux transformations without explicit expressions. Now we compute the more explicit expression of the Darboux matrix of degree two. 131). 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 .

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). 241) h∗j hk = 0 holds at one point (x0 , t0 ). 241) holds at one point, then it holds everywhere. This is proved as follows. ¯ j , hence When λj = λk , λk = −λ hk,x = U (λk )hk , hk,t = V (λk )hk , ∗ ∗ ∗ ¯ j ) = −h∗ U (λk ), hj,x = hj U (λj ) = −h∗j U (−λ j ∗ ∗ ∗ ∗ ∗ ¯ hj,t = hj V (λj ) = −hj V (−λj ) = −hj V (λk ). This implies that (h∗j hk )x = 0, (h∗j hk )t = 0.