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A geometric characterization of the nonlinear Schrödinger equation. (English) Zbl 1099.37055
Summary: We prove that the nonlinear Schrödinger equation of attractive type (NLS\(^+)\) describes just spherical surfaces (SS) and the nonlinear Schrödinger equation of repulsive type (NLS\(^-)\) determines only pseudo-spherical surfaces (PSS). This implies that though we show that given two differential PSS (resp. SS) equations there exists a local gauge transformation (despite of changing the independent variables or not) which transforms a solution of one into any solution of the other, it is impossible to have such a gauge transformation between the NLS\(^+\) and the NLS\(^-\).

MSC:
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
35Q55 NLS equations (nonlinear Schrödinger equations)
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