Solitary wave propagation in a two-dimensional lattice

We construct a family of exact planar solitary wave solutions in a two-dimensional lattice. The system under consideration is a scalar two-dimensional extension of a nonintegrable Fermi-Pasta-Ulam problem with a piecewise quadratic potential. The constructed solutions exhibit an anisotropic dependence on the angle of propagation. Through a detailed analysis of explicit solutions, we show that conventional quasicontinuum models fail to fully describe this dependence. However, a truncated series approximation of the constructed solution that includes sufficiently short wavelengths captures this effect quite well.