×

zbMATH — the first resource for mathematics

Gauge equivalence of differential equations describing surfaces of constant Gaussian curvature. (English) Zbl 1027.53005
The authors of this interesting paper generalize the result of N. Kamran and K. Tenenblat [J. Differ. Equ. 115, 75-98 (1995; Zbl 0815.35036)] to differential equations or systems describing surfaces of constant Gaussian curvature and interprete them geometrically in terms of local gauge transformations. This reveals a quite universal phenomenon that local gauge transformations always exist between differential equations or systems describing pseudospherical surfaces (resp. spherical surfaces). The local gauge transformation transforms a generic solution of one into any generic solution of the other.
The theory is applicable to some known equations: Nonlinear Schrödinger equations (NLS\(^+\) and NLS\(^-\)), \(iq_t+q_{xx}\pm 2|q|^2q=0\), the HF model, that is, the Schrödinger flow of maps into \(S^2\hookrightarrow \mathbb R^3\), the Landau-Lifschitz equation for an isotropic chain \(\mathbf S_t=S\times S_{xx}\) and etc.

MSC:
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35Q58 Other completely integrable PDE (MSC2000)
35A25 Other special methods applied to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
PDF BibTeX XML Cite