Exactly Solvable Ginzburg-Landau theories of Superconducting Order Parameters coupled to Elastic Modes

We consider two families of exactly solvable models describing thermal fluctuations in two-dimensional superconductors coupled to phonons living in an insulating layer, and study the stability of the superconducting state with respect to vortices. The two families are characterized by one or two superconducting planes. The results suggest that the effective critical temperature increases with the thickness of the insulating layer. Also the presence of the additional superconducting layer has the same effect.