The existence of heteroclinic travelling waves in the discrete sine-Gordon equation with nonlinear interaction on a $\mathrm{2D}$-lattice

The article deals with the discrete sine-Gordon equation that describes an infinite system of nonlinearly coupled nonlinear oscillators on a $\mathrm{2D}$-lattice with the external potential $V(r)=K(1-\cos r)$. The main result concerns the existence of heteroclinic travelling waves solutions. Sufficient conditions for the existence of these solutions are obtained by using the critical points method and concentration-compactness principle.