C∞C∞‐Structures for Liénard Equations and New Exact Solutions to a Class of Klein–Gordon Equations

Liénard equations are analyzed using the recent theory of ‐structures. For each Liénard equation, a ‐structure is determined by using a Lie point symmetry and a ‐symmetry. Based on this approach, a novel method for integrating these equations is proposed, which consists in solving sequentially two completely integrable Pfaffian equations. The first Pfaffian equation can be solved by quadrature. For a class of Liénard equations admitting a certain ‐symmetry, this primitive is explicitly determined. The integration procedure is then completed by solving a one‐parameter family of first‐order equations. Additionally, for some subclasses of Liénard equations, an extra one‐parameter family of solutions can be obtained via quadrature. These results are used to derive new exact solutions for a nonlinear Klein–Gordon equation with variable coefficients.