On integrability of Weingarten surfaces: a forgotten class

Abstract

Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equationpossesses a zero curvature representation, a third-order symmetry, and a nonlocal transformation to the sine-Gordon equation φ ξη = sin φ. We leave open the problem of finding a Bäcklund autotransformation and a recursion operator that would produce a local hierarchy.

Computing with Hamiltonian operators

“…2. The Maple package Jets [11], initially developed by Marvan (2003), then also by Baran (2010), has been used for computing Hamiltonian operators for particular differential equations [9,8,10,12]. However, it does not contain any specific feature for integrability structures (like an implementation of the Schouten bracket).…”

Geometry of jet spaces and integrable systems

“…associated with an integrable class of Weingarten surfaces [109] is bi-Hamiltonian with operators D 2…”

More exact solutions of the constant astigmatism equation

“…In the paper, the surfaces were given a name and Equation (1) was derived for the first time. The geometric connection between surfaces of constant astigmatism and pseudospherical surfaces provides a transformation [2,Sect. 6 and 7] between the CAE and the famous sine-Gordon equation…”

“…• Von Lilienthal solutions z = c − y 2 and z = 1/(c − x 2 ), where c is a constant. For details and corresponding surfaces of constant astigmatism see [2,13,16].…”