Kink - Antikink Interactions in the Double Sine - Gordon Equation

As 71 is varied over the range ~ < ~/< ~c, this potential exhibits a variety of structures. Although in later sections we shall discuss these different structures in some detail [2], it is important here to provide a qualitative feeling for them. Clearly, for ~ _+ oo, 'V(~) reduces to a sine-Gordon [SG] form for q~, while for 71 = 0, it leads to a SG form for q~/2. Thus in both these limits the familiar sine-Gordon soli tons-kinks and breathers-will occur as spatially localized, nonlinear excitations in the physical systems being modelled. The behavior of V(q~) for general ~/ can usefully be divided into four regions [2] as shown in figs. l a d . In region I, defined by ~ < 1⁄4 (fig. la), the potential has two distinct types of minima, degenerate in energy and separated by inequivalent barriers. Recalling that topological "kink" solitons connect adjacent degenerate minima of the potential, we see that there are two different types of kinks in this region. In regions II (fig. lb) and II'