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