Local Bifurcations and Traveling Waves of a Periodic Boundary-Value Problem for the Benney–Lin–Kawahara Equation

A periodic boundary value problem for a nonlinear partial differential equation is considered. This equation may be called the Benney–Lin–Kawahara equation as it is its natural generalization. For this periodic boundary value problem, issues related to the existence and stability of traveling-wave solutions are investigated. Asymptotic formulas are also obtained for such solutions. It is shown that, for the boundary value problem under study, these solutions form two-dimensional invariant manifolds that can be interpreted as “packets” of traveling waves. Stability of two-dimensional invariant manifolds, as well as the stability of the solutions that form them, are studied. In particular, it is shown that such solutions are always unstable in the sense of Lyapunov’s definition. However, when these solutions belong to local attractors of the boundary value problem, they are stable in the sense of Perron, Poincaré, and Zhukovsky.