Transformations of ordinary differential equations via Darboux transformation technique

Abstract

Abstract. A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of the solutions of the overdetermined linear systems are derived in the frameworks of the Darboux transformations technique.

Coalescence, deformation and Bäcklund symmetries of Painlevé IV and II equations

“…For example the fourth Painlevé equation, to which we will refer as P IV , exhibits symmetry under Bäcklund transformations that form the affine Weyl group of type A (1) 2 and the second Painlevé equation, to which we will refer as P II , is invariant under Bäcklund transformations from the affine Weyl group A (1) 1 . Bäcklund transformations have also been extensively studied in connection with the Schlesinger transformations, see for instance references [10,15,22] for the case of Painleve II and IV equations. Hybrid Painlevé equations have been a focus of several papers, e.g.…”

Coalescence, Deformation and Bäcklund Symmetries of Painlevé IV and II Equations

“…1 . Bäcklund transformations have also been extensively studied in connection with the Schlesinger transformations, see for instance references [10,15,23] for the case of Painleve II and IV equations.…”