By Frederick A. Valentine
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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. For aj (j ≤ n), this has been proved for the AKNS system; for aj (j ≥ n + 1), the conclusion follows from the boundary condition at infinity. Therefore, the following theorem holds. 192) for λ = λ0 . 192). 200) to the solution p = p + 2λ0 sin θ of the same equation. Moreover, u = −u + 2θ with suitable boundary condition, where p = −ux /2, p = −ux /2.
Remark 15 For the sine-Gordon equation, the Backlund ¨ transformation is a kind of method to get explicit solutions, which was known in the nineteenth century. In that method, to obtain a new solution from a known solution, there is an integrable system of differential equations to be solved (moreover, one can obtain explicit expression by using the theorem of permutability and the nonlinear superposition formula). Using Darboux transformation, that explicit expression can be obtained directly. This will be discussed in Chapter 4 together with the related geometric problems.
Denote Pij be the entries of P , Pij be 25 1+1 dimensional integrable systems the αth derivative of Pij with respect to x. Suppose the order of the highest derivatives of P in ∆diag k+1 is r, then ∂∆diag k+1 = 0= ∂x r ∂∆diag k+1 (α) Pij i,j α=0 ∂P (α+1) Pij . 110) (r+1) In this equation, the coefficient of Pij should be 0. Hence ∆diag k+1 does not contain the rth derivative of P , which means that it is independent of P . 89) implies ∆diag k+1 = 0. 72) in the AKNS system, the Darboux transformation transforms a solution of an equation to a new solution of the same equation.