Nonlocal topological solitons of the sine-Gordon equation

A special nonlocal symmetry related to the Bcklund transformation is obtained for the (1+1)-dimensional sine-Gordon (sG) equation in its polynomial form, which can further generate infinitely many nonlocal symmetries of the sG equation due to the inclusion of an arbitrary spectrum parameter. This nonlocal symmetry can be localized, and hence turn into a point symmetry for the augmented system through the introduction of an auxiliary function. Consequently, finite transformation, similarity reductions, and explicit invariant solutions related to the nonlocal symmetry are obtained. It is known that the sG equation has topological solitons or 2kπ kinks (or antikinks). Based on the group invariant solutions, a different kind of topological soliton, which is a topological soliton situated on a cnoidal wave background, is obtained analytically and graphically displayed.