The (2+1)-dimensional sine-Gordon equation; Integrability and localized solutions

In this paper, the (2 + 1)-dimensional sine-Gordon equation (2DSG) introduced by Konopelchenko and Rogers is investigated and is shown to satisfy the Painleve property. A variable coefficient Hirota bilinear form is constructed by judiciously using the Painleve analysis with a non-conventional choice of the vacuum solutions. First the line kinks are constructed. Then, exact localized coherent structures in the 2DSGI equation are generated by the collision of two non-parallel ghost solitons, which drive the two non-trivial boundaries present in the system. Also the reason for the difficulty in identifying localized solutions in the 2DSGII equation is indicated. We also highlight the significance of the asymptotic values of the boundaries of the system.