On Differential Systems Describing Surfaces of Constant Curvature

Abstract

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 S 2 (the HF model) and its generalized version, the Landau–Lifschitz equation, are shown to describe spherical surfaces. The Schrödinger flow of maps into H 2 (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 u t=−v xx+H 11(u,v)u x+H 12(u,v)v x+H 13(u,v), v t=−u xx+H 21(u,v)u x+H 22(u,v)v x+H 23(u,v), 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 iq t + q xx ± iγ(∣ q∣ 2 q) x − iαq x ± σ∣ q∣ 2 q− 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.