A note on new solitary and similarity class of solutions of a fourth-order non-linear evolution equation

In this paper, Lie’s method is used to calculate new class of solutions of a less studied fourth-order non-linear partial differential equation. In our previous paper [A. Huber, Appl. Math. Comp. 166/2 (2005) 464], we have applied the tanh-method to generate solutions. In this case, special class of solutions in form of traveling waves result (single soliton solutions as well as class of irregular solutions). Therefore, general families of solutions are of basic interest. Doing so, new results can be found in [A. Huber, Appl. Math. Comp., in press] using the Weierstraß transformation derived first by the author. Moreover, a complete characterization of the group properties is given because group properties are unknown up to now. We determine the Lie point symmetry vector fields and calculate similarity “ansätze”. Further, we also derive a few non-linear transformations and some similarity solutions are obtained explicitly. The main purpose for the application of Lie’s method is of course the fact that we are able to calculate class of general solutions which do not underlie such strong restrictions as in the case of traveling wave “ansätze”. Otherwise, it is necessary to perform a group analysis in order to improve the solution manifold by an alternative way. In addition, the studied equation passes the Painlevé-test and seems to be integrable. General class of solutions in terms of elliptic functions are derived via Lie’s approach for the first time representing exact solitary wave propagation and surprisingly, cusp–solitary class of solutions are obtained.