Riemann functions and the group E(1,1)

Abstract

Historically Lie algebras of first-order symmetry operators have proven to be a useful method for finding equivalence classes of Riemann functions. Here this idea is extended to higher order symmetries. The approach is to seek self-adjoint linear hyperbolic partial differential equations that separate variables in more than one coordinate system under the action of the group E(1,1). The equations derived admit no nontrivial first-order operators and can only be obtained from second-order symmetry operators. Using this symmetry structure, a new equivalence class of Riemann functions can then be found.

On the Riemann Function

“…In 2004, Zeitsch [68,69] was able to establish a new equivalence class of Riemann functions by extending the point symmetry ideas of Section 4.1 to Lie-Bäcklund symmetries. The key result was given in [68].…”

“…R 5 is obtained by interchanging the roles of ξ and η in (86)-(88). The Riemann function for (154) first appeared in[69] for the case where δ = ω. The full equation has not been published before.…”