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arXiv · 1002.0989

On integrability of Weingarten surfaces: a forgotten class

Hynek Baran·Michal Marvan
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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 equation $$ z_{yy} + (1/z)_{xx} + 2 = 0 $$ possesses a zero curvature representation, a third-order symmetry, and a nonlocal transformation to the sine-Gordon equation $ϕ_{ξη} = \sinϕ$. We leave open the problem of finding a Backlund autotransformation and a recursion operator that would produce a local hierarchy.

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Hynek Baran, Michal Marvan. 2010-02-04. On integrability of Weingarten surfaces: a forgotten class. https://doi.org/10.1088/1751-8113%2F42%2F40%2F404007

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