Recent attention has focused on deriving localised pulse solutions to various systems of reaction–diffusion equations. In this paper, we consider the evolution of localised pulses in the Brusselator activator–inhibitor model, long considered a paradigm for the study of non-linear equations, in a finite one-dimensional domain with feed of the inhibitor through the boundary and global feed of the activator. We employ the method of matched asymptotic expansions in the limit of small activator diffusivity and small activator and inhibitor feeds. The disparity of diffusion lengths between the activator and inhibitor leads to pulse-type solutions in which the activator is localised while the inhibitor varies on an O(1) length scale. In the asymptotic limit considered, the pulses become spikes described by Dirac delta functions and evolve slowly in time until equilibrium is reached. Such quasi-equilibrium solutions with N activator pulses are constructed and a differential-algebraic system of equations (DAE) is derived, characterising the slow evolution of the locations and the amplitudes of the pulses. We find excellent agreement for the pulse evolution between the asymptotic theory and the results of numerical computations. An algebraic system for the equilibrium pulse amplitudes and locations is derived from the equilibrium points of the DAE system. Both symmetric equilibria, corresponding to a common pulse amplitude, and asymmetric pulse equilibria, for which the pulse amplitudes are different, are constructed. We find that for a positive boundary feed rate, pulse spacing of symmetric equilibria is no longer uniform, and that for sufficiently large boundary flux, pulses at the edges of the pattern may collide with and remain fixed at the boundary. Lastly, stability of the equilibrium solutions is analysed through linearisation of the DAE, which, in contrast to previous approaches, provides a quick way to calculate the small eigenvalues governing weak translation-type instabilities of equilibrium pulse patterns.
- Brusselator model
- pulse evolution
- differential-algebraic equations
- matched asymptotic expansions
- singular perturbations