Exponential Dichotomies for Solitary-Wave Solutions of Semilinear Elliptic Equations on Infinite Cylinders

In applications, solitary-wave solutions of semilinear elliptic equationsΔu+g(u,∇u)=0(x,y)∈R×Ωin infinite cylinders frequently arise as travelling waves of parabolic equations. As such, their bifurcations are an interesting issue. Interpreting elliptic equations on infinite cylinders as dynamical systems inxhas proved very useful. Still, there are major obstacles in obtaining, for instance, bifurcation results similar to those for ordinary differential equations. In this article, persistence and continuation of exponential dichotomies for linear elliptic equations is proved. With this technique at hands, Lyapunov–Schmidt reduction near solitary waves can be applied. As an example, existence of shift dynamics near solitary waves is shown if a perturbationμh(x, u, ∇u) periodic inxis added