Modulational and numerical solutions for the steady discrete Sine-Gordon equation in two space dimensions

We develop a modulation theory based on a suitably averaged Lagrangian for steady solutions of the Sine-Gordon equation in a two dimensional lattice. These lump solutions are nonzero in circular or polygonal regions and zero elsewhere. The modulation theory gives approximate solutions away from small perturbations of the exact anti-continuum solutions for both radial and polygonal solutions. We show how the Peierls-Nabarro potential determines the shape of the boundary between excited sites and the zero solution. These solutions are compared with the corresponding numerical solutions and significant agreement is found. Moreover, we show that solutions with a large radius (more than sixteen lattice sites) can be explained using a continuous trial function for the averaged Lagrangian, while smaller polygonal solutions can be constructed using a trial function, which takes into account the angular variation of the boundary imposed by the lattice. Finally, the ideas of equivariant bifurcation theory are used to obtain a full numerical description of the solution branches as functions of the coupling parameter between neighboring sites. The results of this work can be used to study steady solutions for other types of lattice equations.