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On preliminary symmetry classification of nonlinear Schrödinger equations with some applications to Doebner-Goldin models. (English) Zbl 0970.81024
Summary: We perform classification of a class of one-dimensional nonlinear Schrödinger equations whose symmetry groups have dimensions $$n=1,2, 3$$. Next, from so constructed classes of invariant equations we select those nonlinear Schrödinger equations which are invariant with respect to the Galilei group and its natural extensions. The results obtained are applied for the symmetry classification of complex Galilei-invariant Doebner-Goldin models.

##### MSC:
 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35Q55 NLS equations (nonlinear Schrödinger equations) 58J70 Invariance and symmetry properties for PDEs on manifolds
##### Keywords:
Galilei-invariant
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##### References:
 [1] Białynicki-Birula, I.; Mycielski, J., Ann. phys., 100, 62-74, (1976), (N.Y.) [2] Doebner, H.-D.; Goldin, G.A., Phys. lett. A, 162, 397-401, (1992) [3] Doebner, H.-D.; Goldin, G.A., J. phys. A: math. gen., 27, 1771-1780, (1994) [4] Kibble, T.W.B., Commun. math. phys., 64, 73-82, (1978) [5] Guerra, F.; Pusterla, M., Lett. nuovo cim., 34, 351-356, (1982) [6] Smolin, L., Phys. lett. A, 113, 408-412, (1986) [7] Vigier, J.-P., Phys. lett. A., 135, 99-105, (1989) [8] Sabatier, P.C., Inverse problems, 6, L47-L53, (1990) [9] Bertolami, O., Phys. lett. A, 154, 225-229, (1991) [10] Nattermann, P., Rep. math. phys., 36, 387-396, (1995) [11] Nattermann, P.; Zhdanov, R.Z., J. phys. A: math. gen., 29, 2869-2886, (1996) [12] Ovsjannikov, L.V., Group analysis of differential equations, (1982), Academic Press New York [13] Olver, P.J., Applications of Lie groups to differential equations, (1996), Springer Berlin [14] Jacobson, N., Lie algebras, (1979), Dover Publications New York · JFM 61.1044.02 [15] Mubarakzyanov, G.M., Izvestiya vysshykh uchebnykh zavedenij. matematika, 32, 114-123, (1963) [16] Mubarakzyanov, G.M., Izvestiya vysshykh uchebnykh zavedenij. matematika, 34, 99-106, (1963) [17] Turkowski, P., J. math. phys., 29, 2139-2144, (1988) [18] R. Z. Zhdanov and V. I. Lahno: Group classification of heat conductivity equations with a nonlinear source, math-ph/9906003. · Zbl 0990.35009 [19] Akhatov, I.S.; Gazizov, R.K.; Ibragimov, N.K., (), 3-83 [20] Rideau, G.; Winternitz, P., J. math. phys., 34, 558-570, (1993) [21] Zhdanov, R.Z.; Fushchych, W.I., J. non. math. phys., 4, 426-435, (1997) [22] Fushchych, W.I.; Lahno, V.I., Ukrain. math. J., 51, 341-349, (1999) [23] Güngör, F., J. phys. A: math. gen., 32, 977-988, (1999) [24] Niederer, U., Helv. phys. acta, 45, 802-810, (1972)
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