Bäcklund Transformations of MKdV and Painlevé Equations
نویسنده
چکیده
For N ≥ 3 there are SN and DN actions on the space of solutions of the first nontrivial equation in the SL(N) MKdV hierarchy, generalizing the two Z2 actions on the space of solutions of the standard MKdV equation. These actions survive scaling reduction, and give rise to transformation groups for certain (systems of) ODEs, including the second, fourth and fifth Painlevé equations. Given a solution j of the MKdV equation jt = jxxx − 3 2 j jx (1) we can construct new solutions, −j and j − 2 q , where q satisfies qx + qj = 1 qt + q(jxx − 1 2 j ) = (jx − 1 2 j ). (2) Equations (2) constitute a strong auto-Bäcklund transformation for the MKdV equation, distinct from the usual one given in the literature (see for example [1], Chapter 8, Exercise 2), and discovered, I believe, in the context of Painlevé analysis [2]. If we choose the integration constant arising in the solution of (2) appropriately, the square of this transformation is the identity; but when combined with the j → −j transformation it can,
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