Periodic Hunter-Saxton equation parametrized by the speed of the Galilean frame: Its new solutions, Nucci?s reduction, first integrals and Lie symmetry reduction

The main purpose of this study is to introduce symmetry analysis and Nucci's reduction method for revealing the exact solutions of the periodic Hunter-Suxon equation, which is parameterized by the speed of the Galilean frame. Under one-point similarity transformations, we obtained a set of generators that transformed the given equation into an ordinary differential equation, which can be conveniently taken into consideration for exact solutions. Under special choices, we got three separate families of vector fields out of the problem and constructed solutions for every family. In each case, the Nucci reduction technique generated a solution with a first integral. 3D, density, and contour graphics are given for a better understanding of the solutions.