Lie symmetries and reductions via invariant solutions of general short pulse equation

Abstract

Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) under one-parameter Lie group of transformation to find the invariant, similarity, or auto-model solutions. Lie symmetry analysis (LSA) provides us an algorithm to search for point symmetries for solving related linear systems for infinitesimal generators. Actually, point symmetries lead us to one-parameter family of solutions from a known solution. LSA is a program that provides us the exact solutions for the non-linear differential equations (DEs) in analogy of the program designed by Galois for algebraic polynomial equations. In this paper, we have carried out the LSA for computing the similarity solutions (symmetries) of the non-linear short pulse equation (SPE) for the cases when h(u) = eu, k(u) = uxx, h(u)=eun, and k(u) = uxx. In addition, an optimal system of one-dimensional sub-algebra has been set up. The reductions and invariant solutions for the generalized SPE are calculated corresponding to this optimal system as well. Reductions reduce the non-linear PDE or system of PDEs into non-linear reduced ordered ODE or system of PDEs. This helps to solve these systems of PDEs to reduced form. Graphical behavior of the transformed points of the 1-parameter solution functions have drawn.