| Abstract: |
| We present results concerning the justification of modulation equations for the dynamics of atoms in a two-dimensional square lattice that interact with their nearest neighbors nonlinearly with respect to the strain between their scalar displacements. When the interaction forces are cubic we show that small macroscopically modulated amplitudes of rapidly oscillating plane waves evolve approximately according to a nonlinear Schroedinger equation, while in the case of quadratic interaction forces the corresponding modulation equation is a Davey-Stewartson system. Due to the dispersive scaling of the small amplitudes, the justification in the latter case is significantly more involved than in the former and necessitates the employment of normal-form transformation techniques. |
|