Nonclassical potential symmetry analysis and exact solutions for a thin film model of a perfectly soluble anti-surfactant solution

Abstract

In this article, we construct exact solutions to a quasilinear system of hyperbolic partial differential equations which governs the dynamics of a thin film of a perfectly soluble anti-surfactant solution. The symmetry analysis for this system is performed first time in the literature to the best of our knowledge. In fact, we present a detailed and comprehensive study of symmetry analysis for the governing system and compute classical symmetries, nonclassical symmetries, nonlocal symmetries and nonclassical potential symmetries. Nonclassical potential symmetries appear to be very much useful in terms of obtaining several new hidden solutions of the system that can not be established by using classical symmetry reductions, nonclassical symmetry method, or potential symmetry analysis. By using the direct multipliers we demonstrate several conserved quantities of the governing system those yield associated nonlocally related potential systems to the given system. Further, we prove that the given system admits a nonlocal symmetry that arises from the symmetry-based method and consequently we obtain a family of exact solutions. We analyze the physical interpretation of some of the obtained solutions which include various soliton type solutions such as kink type solitons, breather type solitons, multiple solitons, singular kink type solitons and etc. Lastly, as an application, we analyze the evolutionary property of characteristic shock, weak discontinuity and collision between them by using one of the obtained solutions.