1998/05/04 by E. V. Ferapontov, Ferapontov, E. V.
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations #math.DG
paper · pdf · doi:10.48550/arxiv.math/9805011
Latex, 13 pages
arxiv created 1998/05/04 · openalex publication_date 1998/05/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is demonstrated that the stationary Veselov-Novikov (VN) and the stationary modified Veselov-Novikov (mVN) equations describe one and the same class of surfaces in projective differential geometry: the so-called isothermally asymptotic surfaces, examples of which include arbitrary quadrics and cubics, quartics of Kummer, projective transforms of affine spheres and rotation surfaces. The stationary mVN equation arises in the Wilczynski approach and plays the role of the projective "Gauss-Codazzi" equations, while the stationary VN equation follows from the Lelieuvre representation of surfaces in 3-space. This implies an explicit Backlund transformation between the stationary VN and mVN equations which is an analog of the Miura transformation between their (1+1)-dimensional limits.