Perturbation theories for sine-Gordon soliton dynamics (Q793235)

In this paper three different approximate solutions to the nonlinear sine-Gordon equation are given and compared by numerical solution of the equations. The equation considered in all three cases is \[ \phi_{xx}- \phi_{tt}-\sin \phi =\alpha \phi_ t-\beta \phi_{xxt}+\gamma +\mu \delta(x)\sin \phi, \] where \(\alpha\),\(\beta\),\(\gamma\) are constants and \(\delta\) (x) is the usual \(\delta\)-function. Three different forms for \(\phi\) are assumed based on certain physical assumptions which lead to three corresponding different first-order ordinary differential equations. These are integrated and it is found that the previously formulated McLaughlin-Scott theory provides the closest of these three approximations to the answer.