Dilaton domain walls and dynamical systems

Domain wall solutions of d-dimensional gravity coupled to a dilaton field σ with an exponential potential Λ e-λσ are shown to be governed by an autonomous dynamical system, with a transcritical bifurcation as a function of the parameter λ when Λ < 0. All phase-plane trajectories are found exactly for λ = 0, including separatrices corresponding to walls that interpolate between AdSd and AdSd-1 x R, and the exact solution is found for d = 3. Janus-type solutions are interpreted as marginal bound states of these 'separatrix walls'. All flat domain wall solutions, which are given exactly for any λ, are shown to be supersymmetric for some superpotential W, determined by the solution