A note on the elliptic sine-Gordon equation

The elliptic sine-Gordon equation originates from the static case of the hyperbolic sine-Gordon equation modeling the Josephson junction in superconductivity. However, the elliptic sine-Gordon boundary value problem as studied in the mathematical literature actually has an opposite sign in front of the sine nonlinearity; it models not the “usual” Josephson junction but rather the Josephson π-junction, which is of contemporary interest to physicists. We first furnish this physical backdrop that has motivated our study here. Then we aim to establish the existence of nonconstant solutions of the semilinear elliptic sine-Gordon equation subject to homogeneous Neumann and Dirichlet boundary conditions by using critical point theory. Positive numerical solutions of the Dirichlet case, which are global minima of the variational problem, are computed on a dumbbell-shaped 2D domain for visualization. 1. Origin of the model The hyperbolic sine-Gordon equation (1.1) φxx − φtt = sin φ describes the dynamics of many condensed matter systems. Examples include: a 1-D ferromagnet with planar anisotropy in the presence of a magnetic field perpendicular to the chain direction [Mi], spin dynamics of the A-phase of superfluid He [MK], and a Josephson transmission line [Sc], etc., besides such classical examples as: a chain of coupled pendula [Dr], and a classical model on 1-D dislocation [La]. The sine-Gordon equation may be derived from the Lagrangian: (1.2) L = ∫ dx ( 1 2 φt − 1 2 φx + cosφ − 1 ) , 2000 Mathematics Subject Classification. Primary: 35J20, 58E05; Secondary: 35Q20, 65N30.