Sine-Gordon Field with Spatial Boundary Condition and Application to Josephson Junction

The two dimensional (one space-one time) sine-Gordon field with a spatial boundary condition is studied. An approximation used here is similar to the weak-coupling approxima tion in a nonlinear field theory, where the Lagrangian density is expanded about the static classical value l' [ If!c'(x)] in powers of r/;(x, t) -If!c,(x). By using If!c'(x), we discuss the magneti zation of the Josephson junction with a finite linear length, taking a spatial boundary condition into account. We find that the magnetization shows not only discontinuities but also para magnetic regions as a function of an applied magnetic field.