On a Nonlinear Third-order Evolution Equation - Present Developments

In this paper the classical Lie group formalism is applied to deduce new classes of solutions of a less studied nonlinear partial differential equation (nPDE) of the third order. The nPDE under consideration is closely related to motions of plane curves. Up to now no carefully performed symmetry analysis is available. Therefore we determine the classical Lie point symmetries including algebraic properties. Similarity solutions are given as well as nonlinear transformations could derived and periodic wave trains are obtained. Since algebraic solution techniques fail symmetry analysis justifies the application yielding a deeper insight into the solution-manifold. In addition, we shall see that the nPDE admits a new symmetry, the so called potential symmetry. For some nonlinear ordinary differential equations (nODEs) created by similarity reductions the Painlevé property is discussed.