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On differential systems describing surfaces of constant curvature. (English) Zbl 1022.58005

Author’s Summary: The geometric notion of a differential system describing surfaces of constant nonzero Gaussian curvature is introduced. The nonlinear Schrödinger equation (NLS) with \(k= 1\) and \(-1\) is shown to describe a family of spherical surfaces and pseudospherical surfaces, respectively. The Schrödinger flow of maps into \(S^2\) and its generalized version, the Landau-Lifschitz equation, are shown to describe spherical surfaces. The Schrödinger flow of maps into \(H^2\) provides another example of a system describing pseudospherical surfaces. New different 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 \(\eta\)-pseudospherical or \(\eta\)-spherical surfaces. As an application, we obtain a four-parameter family of such systems for a complex-valued function \(q= u+iv\) given by \(iq_t+ q_{xx}\pm i\gamma(|q|^2 q)_x- i\alpha q_x\pm \sigma|q|^2 q-\beta q= 0\), where \(\sigma\geq 0\) if \(\gamma= 0\). Particular cases of this family, obtained by the vanishing of the parameters, are the linear equations, the NLS equation, the derivative nonlinear Schrödinger equations (DNLS) and the mixed NLS-DNLS equation.
Reviewer: X.Geng (Henan)

MSC:

58D25 Equations in function spaces; evolution equations
35Q99 Partial differential equations of mathematical physics and other areas of application
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