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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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##### References:
 [1] Tingley, D., Isometries of the unit sphere, Geometriae Dedicata, 1987, 22: 371–378. · Zbl 0615.51005 · doi:10.1007/BF00147942 [2] Ma Yumei, Isometries of the unit sphere, Acta Math. Sci., 1992, 12: 366–373. · Zbl 0808.46018 [3] Ding Guanggui, On the extension of isometries between unit spheres ofE andC($$\Omega$$), Acta Math. Sinica (New Series), 2001, 17: 1. · Zbl 0980.42036 · doi:10.1007/s101140000092 [4] Rassias, T. M., Properties of isometries and approximate isometries, Recent Progress in Inequalities, MIA, 325–345, Drichlet: Kluwer Academic Publishers, 1998. [5] Rassias, Th. M., Semrl, P., On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mapping, Proc. Amer. Math. Soc, 1993, 118: 919–925. · Zbl 0780.51010 · doi:10.1090/S0002-9939-1993-1111437-6 [6] Rolewicz, S., Metric Linear Spaces, Dordrecht-Warszawa: Reidel and PWN, 1985. · Zbl 0573.46001 [7] Baker, J. A., Isometries in normed spaces, Amer. Math. Monthly, 1971, 8: 655–658. · Zbl 0214.12704 · doi:10.2307/2316577 [8] Rudin, W.,Functional Analysis, New York, Toronto: McGraw-Hill Inc.,1973. · Zbl 0253.46001
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