1+1 Dimensional Integrable Systems

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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 .

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. 138) n−j Vj (x, t)λ , j=0 Uj ’s and Vj ’s are N × N matrices.

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. 138) n−j Vj (x, t)λ , j=0 Uj ’s and Vj ’s are N × N matrices.

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