Kink solutions in generalized 2D dilaton gravity

We study static kink solutions in a generalized two-dimensional dilaton gravity model, where the kinetic term of the dilaton is generalized to be an arbitrary function of the canonical one [Formula presented], say F(X), and the kink is generated by a canonical scalar matter field ϕ. It is found that for arbitrary F(X), the background field equations have a simple first-order formalism, and the linear perturbation equation can always be written as a Schrödinger-like equation with factorizable Hamiltonian operator. After choosing appropriate F(X) and superpotential, we obtain a sine-Gordon type kink solution with pure AdS₂ metric. The linear perturbation issue of this solution becomes an exactly solvable conformal quantum mechanics problem, if one of the model parameter takes a critical value.