Exact Solutions to Non-Linear Symmetron Theory: One and Two Mirror Systems

We derive the exact analytical solutions to the symmetron field theory equations in the presence of a one or two mirror system. The one dimensional equations of motion are integrated exactly for both systems and their solutions can be expressed in terms of Jacobi elliptic functions. Surprisingly, in the case of two parallel mirrors the equations of motion generically provide not a unique solution but a discrete set of solutions with increasing number of nodes and energies. The solutions obtained herein can be applied to qBOUNCE experiments, neutron interferometry and for the calculation of the symmetron field induced "Casimir force" in the CANNEX experiment.