Spectral stability of modulated travelling waves bifurcating near essential instabilities

Localized travelling waves to reaction-diffusion systems on the real line a re investigated. The issue addressed in this work is the transition to inst ability which arises when the essential spectrum crosses the imaginary-axis . In the first part of this work, it has been shown that large modulated pu lses bifurcate near the onset of instability; they are a superposition of t he primary pulse with spatially periodic Turing patterns of small amplitude . The bifurcating modulated pulses can be parametrized by the wavelength of the Turing patterns. Furthermore, they are time periodic in a moving frame . In the second part, spectral stability of the bifurcating modulated pulse s is addressed. It is shown that the modulated pulses are spectrally stable if and only if the small Turing patterns are spectrally stable, that is, i f their continuous spectrum only touches the imaginary-axis at zero. This r equires an investigation of the period map associated with the time-periodi c modulated pulses.