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The linear stability of the travelling wave solutions of a general reaction-diusion system is investigated. The spectrum of the corresponding second order dieren tial operator L is studied. The problem is reduced to an asymptotically autonomous rst order linear system. The relation between the spectrum of L and the corresponding rst order system is dealt with in detail. The rst order system is investigated using exponential dichotomies. A self-contained short presentation is shown for the study of the spectrum, with elementary proofs. An algorithm is given for the determination of the exact position of the essential spectrum. The Evans function method for determining the isolated eigenvalues of L is also presented. The theory is illustrated by three examples: a single travelling wave equation, a three variable combustion model and the generalized KdV equation.

On the structure of spectra of travelling waves