A note on a class of solitary-like solutions of the Tzitzéica equation generated by a similarity reduction

In this paper, new class of solutions of the Tzitzéica equation are derived by using the classical Lie symmetry analysis. The important aspect of this paper however is the fact that the analysis results in a new class of solitary-like solutions, the so-called cusp-solitary solutions.This special type of solutions are not found in the current literature and represents a necessary contribution for the whole solution manifold. The studied equation was originally found in the field of geometry, otherwise the Tzitzéica equation takes place in many branches of non-linear sciences. Therefore, explicit class of solutions connected by a physical meaning are of great importance. The analysis is restricted to the case of traveling waves represented by a similarity variable describing any wave propagation. A complete characterization of the group properties is given. The classical Lie point symmetries are derived and algebraic properties are determined. Similarity solutions and transformations are given in a most general form and have been derived for the first time in terms of Jacobian elliptic functions. It is worth to mention that the application of known powerful algebraic methods (e.g. special function transform methods) are not appropriate to study the solution manifold. Hence, the present paper is therefore suitable to create a deeper insight into the solution manifold with respect to the traveling wave picture.