Kink propagation and trapping in a two dimensional curved sine-Gordon system

The sine-Gordon equation is one of the three classical nonlinear partial differential equations possessing soliton solutions in the case of one spatial dimension. Extending the sine-Gordon equation to two spatial dimensions is relevant for applications to the dynamics of large area Josephson junctions.

In particular we have investigated Josephson waveguides consisting of two rectangular large area regions joined by a bent section of constant curvature. Transverse homogeneous and inhomogeneous Neumann boundary conditions are used. Numerical and approximate analytical tools have been used to investigate how kink shaped solitons of the sine-Gordon equation propagate through the bent section.

We have found that the region with finite curvature acts as a potential barrier whose height and width depends on the radius of curvature of the waveguide. The kink transmission, reflection and trapping is investigated. The kink may be captured when a driving force, provided by a magnetic field, is applied to the kink.

The approximate analytical tools are based on a variational approach of a Lagrangian. The variation is done with respect to collective coordinates, or slowly varying coefficients, in the 1D soliton solution.