Evolution equations and \(\text{sl}(2, \mathbb R)\)-valued Bäcklund forms (Q1089842)

The main concepts of the paper ascend to the works of \textit{H. D. Wahlquist} and \textit{F. B. Estabrook} [J. Math. Phys. 16, 1--7 (1975; Zbl 0298.35012)] where relations were discovered between the so called Bäcklund forms on jet bundles and partial differential equations. The present paper contains a classification of those \(\text{sl}(2, \mathbb R)\)-valued Bäcklund forms (on a jet bundle with one \(t\) and one \(x\)) which give rise to evolution equations. KdV, MKdV and generalized sine-Gordon equations appear in a natural way in the course of the classification. A geometrical setting is further given for Gardner's construction [see \textit{M. Miura}, SIAM Rev. 18, 412--459 (1976; Zbl 0333.35021)] of conserved quantities.