Exact solutions and critical chaos in dilaton gravity with a boundary

We consider (1+1)-dimensional dilaton gravity with reflecting
dynamical boundary. The boundary cuts off the region of strong
coupling and makes the model causally similar to the multidimensional
spherically-symmetric gravity. We
demonstrate that this model is exactly solvable at the classical level
and possesses an on-shell SL(2,R) symmetry. After introducing general
classical solution, we study in detail a large class of particular
solutions describing reflection of matter waves off the boundary at
low energies and formation of black holes at energies above critical. 
We argue that  despite being exactly solvable, the model in the
critical regime, i.e. at the verge of black hole formation, displays
dynamical instabilities specific to chaotic systems.