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 th e periodic wave trains given that the pulse is stable. Using the Evans func tion, Gardner has demonstrated that every isolated eigenvalue of the linear ization about the pulse generates a small circle of eigenvalues for the lin earization about the periodic waves. In this article, the precise location of these circles is determined. It is demonstrated that the stability prope rties of the periodic waves depend on certain decay and oscillation propert ies of the tails of the pulse. As a consequence, periodic waves with long w avelength typically destabilize at homoclinic bifurcation points at which m ultihump pulses are created. That is in contrast to the situation for the u nderlying pulses whose stability properties are not affected by these bifur cations. The proof uses Lyapunov-Schmidt reduction and relies on the existe nce of exponential dichotomies. The approach is also applicable to periodic waves with large spatial period of elliptic problems on R " or on unbounde d cylinders R X Omega with Omega bounded. (C) 2001 Academic Press.