On Differential Systems Describing Surfaces of Constant Curvature

The geometric notion of a differential system describing surfaces of constant nonzero Gaussian curvature is introduced. The nonlinear Schrödinger equation (NLS) with κ=1 and −1 is shown to describe a family of spherical surfaces (s.s.) and pseudospherical surfaces (p.s.s.), respectively. The Schrödinger flow of maps into S2 (the HF model) and its generalized version, the Landau–Lifschitz equation, are shown to describe spherical surfaces. The Schrödinger flow of maps into H2 (the M-HF model) provides another example of a system describing pseudo-spherical surfaces. New differential systems describing surfaces of nonzero constant Gaussian curvature are obtained. Furthermore, we give a characterization of evolution systems which describe surfaces of nonzero constant Gaussian curvature. In particular, we determine all differential systems of type which describe η-pseudospherical or η-spherical surfaces. As an application, we obtain four-parameter family of such systems for a complex-valued function q=u+iv given by iqt+qxx±iγ( q 2q)x−iαqx±σ q 2q−βq=0, where σ 0 if γ=0. Particular cases of this family, obtained by the vanishing of the parameters, are the linear equations, the NLS equation, the derivative nonlinear Schrödinger equation (DNLS) and the mixed NLS–DNLS equation.