Dilaton Domain Walls and Dynamical Systems

Domain wall solutions of d-dimensional gravity coupled to a dilaton field σ with an exponential potential Λe 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 × R, and the exact separatrix wall solution is found for d = 3. All flat domain wall solutions, which are given exactly for any λ, are shown to be supersymmetric for some superpotential, determined by the solution, and all supersymmetric non-flat walls are shown to be foliations of anti-de Sitter space.