BOUND-STATES OF THE ELLIPTIC SINE-GORDON EQUATION

We give a detailed study of finite-energy solutions to elliptic sine-G ordon (SG) equation in the plane with point-like singularities, These bound-state solutions (in a sense of scalar field theory) with only on e singularity at the origin demonstrate a target-like annular soliton pattern at large distance from the origin. An effective radius of this pattern is calculated both analytically and numerically for the case of axial symmetric solutions. The analytic study is based on an isomon odromic deformation method for the third Painleve equation, which dist inguishes bound-state solutions as separatrices in a manifold of gener al (infinite-energy) solutions. Exact analytic estimates give us a too l to study bounded-state solutions to the nonintegrable SG equation wi th forcing, Namely, for large intensity at the singularity we derive a critical value of forcing, which governs the existence and stability of the bound-state solutions. This plays a crucial role for two concre te physical applications dealing with large area Josephson junctions a nd nematic liquid crystals in a rotating magnetic field. For both exam ples we compute critical values of field and driving forces which enab les the formation of modes with finite energy. These numerical compute d critical values correlate well with computer simulations and experim ental data.