Bound state solutions of the elliptic sine-Gordon equation

Abstract

We present a detailed investigation of finite-energy solutions with point-like singularities of the elliptic sine-Gordon equation in a plane. Such solutions are of the bound-state type in the sense of scalar field theory. If the solution has a unique singularity, then it behaves as a soliton-like annular wave packet at a large distance from the singularity. The effective radius of this wave packet is evaluated both analytically and numerically for axially symmetric solutions. The analytical investigation is based on the method of isomonodromy deformations for the third Painleve equation, which singles out these solutions as separatrices of the manifold of general solutions (with infinite energy). Exact analytical estimates provide a tool for investigating bound-state solutions of the nonintegrable sine-Gordon equation with a nonzero right-hand side. More precisely, for large-intensity fields at the singularity, we derive the critical forcing that allows the existence and stability of a bound state. As an illustration, we consider two applications: a large-area Josephson junction and a nematic liquid crystal in a rotating magnetic field. For each of the examples, we evaluate the critical values of the field that allow finite-energy regimes. These are in good agreement with numerical and experimental data.