The eigenvalue problem of the general Einstein-Weyl metric equation and exact self-similar and multi-traveling waves solutions

The integrability of the general Einstein-Weyl Metric equation is studied . Here, the approach proposed is based on using the extended and generalized unified methods. These methods inspect the integrability conditions, whenever exist. The exact solutions are thus obtained. They provide the existence and uniqueness of solution for initial value problem where initial value is taken from the exact solutions obtained. Similarity variables are introduced, and the extended unified method is applied to find a class of explicit self-similar wave solutions to the GEWM equation. Numerical evaluations of the solutions are done via symbolic computation. These results show periodic solitons and multi-lumps. It is found that the generalized heavenly equation admits a class of infinite solutions, among them the chaotic ones. The generalized unified method is used to find multi-traveling solitary waves solutions. We think that the presented methods generalize the known ones in the literature. This will be illustrated later on.