Dilaton Domain Walls and Dynamical Systems

Domain wall solutions of $d$-dimensional gravity coupled to a dilaton field $σ$ with an exponential potential $Λe^{-λσ}$ 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 $adS_d$ and $adS_{d-1} \times\bR$, 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 $λ$, are shown to be supersymmetric for some superpotential $W$, determined by the solution.

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