A bstract We report a two-dimensional (2D) gravitating kink model, for which both the background field equations and the linear perturbation equation are exactly solvable. The background solution describes a sine-Gordon kink that interpolating between two asymptotic AdS 2 spaces, and can be regarded as a 2D thick brane world solution. The linear perturbation equation can be recasted into a Schrödinger-like equation with singular Pöschl-Teller II potentials. There is no tachyonic state in the spectrum, so the solution is stable against the linear perturbations. Besides, there can be n = 0 , 1 , 2 , ⋯ bounded vibrational modes around the kink. The number of these vibrational modes varies with model parameters.