Essential instability of pulses and bifurcations to modulated travelling waves

Reaction-diffusion systems on the real line are considered. Localized trave lling waves become unstable when the essential spectrum of the linearizatio n about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a famil y of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcatin g patterns can be parametrized by the wavelength of the steady patterns; th ey are time-periodic in a moving frame. A major difficulty in analysing thi s bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov-Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touchi ng the imaginary axis.