# Darboux Transformations in Integrable Systems. Theory and by Gu C., Hu H., Zhou Z.

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

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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. 171). The solutions of the equations whose coeﬃcients depending on t diﬀer a lot from the solutions of the equations whose coeﬃcients independent of t. In the latter case, each soliton moves in a ﬁxed velocity and the soliton with larger amplitude moves faster. g. oscillates), and the soliton with larger amplitude may move slower.

N r hN r .. 115) which is an N r × N r matrix. 116) Fr = −(λr1 h1 , · · · , λrN r hN r ). 117) j=0 and can be written as This is a system of linear algebraic equations for (Dr , Dr−1 , · · · , D1 ). When det Fr = 0, it has a unique solution (Dr , Dr−1 , · · · , D1 ). Therefore, when det Fr = 0, there exists a unique N ×N matrix D(x, t, λ) satisfying D(x, t, λi )hi = 0 (i = 1, · · · , N r). We write it as D(h1 , · · · , hN r , λ) to indicate that D is constructed from h1 , · · · , hN r .

224) or equivalently, for suitable choice of the integral constant. It remains to prove that the Darboux matrix λI − S keeps the reduction of MKdV-SG hierarchy. This includes (1) the transformed A , B and C still satisfy A (−λ) = −A (λ) and B (−λ) = −C(λ); (2) the coeﬃcients αj (t)’s keeps invariant. Since V T (−λ) = −V (λ), S T = S and (λI + S)T (λI − S) = λ2 I − S 2 = 2 (λ −λ20 )I, it can be veriﬁed by direct calculation that V T (−λ) = −V (λ) holds. This proves (1). (2) is proved as follows.