Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems

AbstractWe consider a spatially homogeneous system of reaction-diffusion equation defined on the interval (−∞, ∞) of the one-dimensional spatial variable x. It is known that this equation has a one-parameter family of periodic travelling wave solutions Ψ(x + ct; c) if this equation has a spatially homogeneous periodic solution φ(t). The spatial period L(c) of the travelling wave solution satisfies L(c)c → T if c → +∞, where c is the propagation speed and T is the period of φ(t). We prove that, in the case c > 0 is sufficiently large, Ψ(x + ct; c) is unstable if φ(t) is “strongly unstable” and Ψ(x + ct; c) is “marginally stable” if φ(t) is “strongly stable.” If the equation is defined on a finite interval [0, l] of the variable x with the periodic boundary conditions, we can obtain a more precise result regarding the stability of Ψ(x + ĉt; ĉ), where ĉ > 0 is a speed which satisfies l = mL(ĉ) for some positive integer m. We prove that this solution is asymptotically stable in the sense of waveform stability if ĉ > 0 is sufficiently large and if φ(t) is “strongly stable.