On integrability of Weingarten surfaces: a forgotten class

Details On integrability of Weingarten surfaces: a forgotten class On integrability of Weingarten surfaces: a forgotten class

2010-02-04

arxiv.org/abs/1002.0989v1

J. Phys. A: Math. Theor. 42 (2009) 404007

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

10.1088/1751-8113/42/40/404007

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 $\phi_{\xi\eta} = \sin\phi$. We leave open the problem of finding a Backlund autotransformation and a recursion operator that would produce a local hierarchy.

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