Multi-dimensional effects on the propagation of strain solitons in solids

Numerical simulations on the generation and the propagation of phonon solitons in crystalline solids have so far been made extensively with KdV equation in one dimension. We extend these analyses to two and three dimensions by treating the effect of the transverse directions as a small perturbation. Explicitly, we first derive the two-dimensional KP equation and also its three-dimensional version for the strain field. Then, time evolutions of the initial strain distributed with a Gaussian profile normal to the main propagation direction are studied