Joerg VOLKMANN and Gerd BAUMANN
Department of Mathematical Physics, University of Ulm,
Albert-Einstein-Allee 11, 89069 Ulm, GERMANYl
E-mail: volk@physik.uni-ulm.de, gerd.baumann@physik.uni-ulm.de

Symmetry analysis of the 2 + 1 dimensional Doebner-Goldin equations

Abstract:
Symmetry-investigations of differential equations are a very powerful tool to examine systems of nonlinear partial differential equations in an analytical way. We calculate for the 2 + 1 dimensional Doebner-Goldin equations the infinitesimals. In an additional step  a reduction of the  independent variables is carried out. These two steps are repeated again and again till an ordinary differential equation results, which will be solved if possible. This method is described in the literature very detailed (for example [1, 2, 3]).

During applications there occur another point of view, which leads to the preliminary group classification. This examination now extend the traditional symmetry investigation of systems of differential equations to a family of differential equations incooperating arbitrary or free functions. The result is a classification of these free functions. This extended examination  leads to equations which are special elements of this family. The equations can be reduced step by step till ordinary differential equations results.

The example deals with the 2 + 1 dimensional Doebner-Goldin equations [4]. After the derivation of the equations, we will consider the \$2 + 1\$ dimensional case. We will calculate the symmetry group and their properties. In the next step we will show some reductions and solutions of these equations.

The second part of our discussion is concerned with the theory of prolangations. We will examine different ways to calculate prolongation formulas of equations containig free functions. A speicial computer algebra programm will be represented. This tool  will be applied to the Doebner-Goldin-equation in \$3 + 1\$ dimensions.

References:

1. G. Baumann, Symmetry Analysis of Differential Equations with {\itshape Mathematica}, Telos, Springer 2000.
2. P.J. Olver, Application of Lie Groups to Differential Equations, Springer, New York 1986.
3. N.H. Ibragimov, Transformation Groups Applied to  Mathematical Physics, D. Reidel Publishing Company, Dordrecht 1985.
4. P. Nattermann, R. Zhdanov, On Integrable Doebner-Goldin Equations, ASI-TPA/12/95