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68) leads to − Ax + C − B(ζ − u) = 0, Bx + 2A = 0, −ut − Cx + 2A(ζ − u) = 0. 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 .

282). Property 6. N and N satisfy the follow system of linear integral equations (Gelfand-Levitan-Marchenko equations) ⎛ N (x, s) + B(2x + s) ⎝ ⎛ N (x, s) + B(2x + s) ⎝ ⎞ 1 0 0 1 +∞ ⎠ + ⎞ N (x, σ)B(2x + s + σ) dσ = 0, 0 +∞ ⎠ + N (x, σ)B(2x + s + σ) dσ = 0. 279) gives (p, q). The process to get scattering data from (p, q) is called the scattering process. It needs to solve the spectral problem of ordinary differential equations. The process to get (p, q) from the scattering data is called the inverse scattering process.

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.

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1+1 Dimensional Integrable Systems

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