Series solutions and bifurcation of traveling waves in the Benney–Kawahara–Lin equation

Abstract

In this paper, the Lie group method is performed on the BKL equation. By reducing the BKL equation to four classes of ODEs, we obtain new power series solutions of it. Furthermore, as an important reduced equation, the traveling wave equation is investigated in detail. Treating it as a singular perturbation system in $${\mathbb {R}}^4$$ , we apply the geometric singular perturbation theory and dynamical system methods to study its phase space geometry. By using some techniques, such as tracking the unstable manifold of the saddle, discussing transversality of the stable and unstable manifolds and investigating some complicated nonlocal bifurcation, we obtain wave speed conditions to guarantee the existence of various bounded traveling waves of the BKL equation.