Dilaton Domain Walls and Dynamical Systems

Domain wall solutions of $d$-dimensional gravity coupled to a dilaton field $\sigma$ with an exponential potential $\mathrm{\Lambda }{e}^{-\lambda \sigma }$ are shown to be governed by an autonomous dynamical system, with a transcritical bifurcation as a function of the parameter $\lambda$ when $\mathrm{\Lambda }<0$. All phase-plane trajectories are found exactly for $\lambda =0$, including separatrices corresponding to walls that interpolate between $ad{S}_d$ and $ad{S}_{d-1}\times \text{bR}$, 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 $\lambda$, are shown to be supersymmetric for some superpotential $W$, determined by the solution.