On the Stability of Periodic Travelling Waves with Large Spatial Period

In many circumstances, a pulse to a partial differential equation (PDE) on the real line is accompanied by periodic wave trains that have arbitrarily large period. It is then interesting to investigate the PDE stability of the periodic wave trains given that the pulse is stable. Using the Evans function, Gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves. In this article, the precise location of these circles is determined. It is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse. As a consequence, periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi-hump pulses are created. That is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations. The proof uses Lyapunov–Schmidt reduction and relies on the existence of exponential dichotomies. The approach is also applicable to periodic waves with large spatial period of elliptic problems on n or on unbounded cylinders ×Ω with Ω bounded.