Solitary waves in two-dimensional nonlinear lattices

Our understanding of nonlinear dynamics critically hinges on rigorous closed-form solutions to nonlinear wave equations; few closed-form solutions have been achieved for physics-based interaction models beyond one-dimensional space. In this paper, we investigate the dynamics of a two-dimensional lattice with harmonic, weakly nonlinear, and strongly nonlinear interactions. Assuming nearest neighbor interaction, we derive the continuum approximation of the discrete system in the long-wavelength regime while keeping the Hamiltonian structure of the system. For a hexagonal lattice with nontrivial shear resistance, we surprisingly find that solitary wave solutions exist in certain directions related to the underlying symmetries of the lattice. The properties of the solitary waves are also studied by numerical simulations of the original discrete system. Besides being of fundamental scientific interest, the solitary wave solutions in nonlinear hexagonal lattices are anticipated to have applications in the design of shock absorbers, acoustic lens, or nondestructive structural testing devices, among many others.