Lie-Bäcklund tangent transformations

Bäcklund Transformations for Integrable Geometric Curve Flows

We study the Bäcklund transformations of integrable geometric curve flows in certain geometries. These curve flows include the KdV and Camassa-Holm flows in the two-dimensional centro-equiaffine geometry, the mKdV and modified Camassa-Holm flows in the two-dimensional Euclidean geometry, the Schrödinger and extended Harry-Dym flows in the three-dimensional Euclidean geometry and the Sawada-Kote...

Tangent Line Transformations

The new curve contains all of the information about the original curve’s tangent lines. For example, it crosses the y-axis whenever the original curve has a horizontal tangent line. The straight line y = mx + b, or (x(t), y(t)) = (t,mt + b), has only one tangent line and thus a trivial transformation to a point, (x∗(t), y∗(t)) = (m, b). Some conic sections transform to conic sections. For examp...

Bäcklund transformations in 2 D dilaton gravity ∗

We give a Bäcklund transformation connecting a generic 2D dilaton gravity theory to a generally covariant free field theory. This transformation provides an explicit canonical transformation relating both theories.

Bäcklund Transformations for Fourth Painlevé Hierarchies Pilar

Bäcklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlevé (PIV ) hierarchies recently found by the same authors [Publ. Res. Inst. Math. Sci. (Kyoto) 37 327–347 (2001)]. We ...

A Characterization of the Infinitesimal Conformal Transformations on Tangent Bundles