Abundant symmetries for the 1+1 dimensional classical Liouville field theory

Abstract

Explicit calculation shows us that there exists a rich associated symmetry structure and abundant strong symmetry configuration for the 1+1 dimensional classical Liouville field theory. Starting from these symmetries and strong symmetries, various integrable hierarchies can be obtained. The Liouville I–III hierarchies, the modified Korteweg–de Vries (KdV) hierarchy, the Fordy–Gibbons I, and Fordy–Gibbons II hierarchies are just special cases while the KdV hierarchy, Caudry–Dodd–Gibbon–Sawada–Kotera hierarchy, and Kaup–Kupershmidt hierarchy can be obtained by using a same Miura transformation from the modified KdV, the Fordy–Gibbons I and Fordy–Gibbons II hierarchies, respectively. Two types of the generalized Riccati hierarchies are also presented.