Lie symmetries of a generalised non-linear Schrodinger equation. III. Reductions to third-order ordinary differential equations

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Abstract
For pt. II see ibid., vol. 22, p.469 (1989). The study of group invariant solutions of the generalised non-linear Schrodinger equation (GNLSE) is continued. It is shown that eight types of subgroups of the symmetry group lead via symmetry reduction, to third-order real ordinary differential equations, giving both the phase and the absolute value of the solution. Only two of the reductions provide to a Painleve type equation and both of them only for the cubic GNLSE. This equation is solved in terms of the fourth Painleve transcendent.