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The amplitude modulation of dust lattice waves (DLWs) propagating in a two-dimensional hexagonal dust crystal is investigated in a continuum approximation, accounting for the effect of dust charge polarization (dressed interactions). A dusty plasma crystalline configuration with constant dust grain charge and mass is considered. The dispersion relation and the group velocity for DLWs are determined for wave propagation in both longitudinal and transverse directions. The reductive perturbation method is used to derive a (2+1)-dimensional nonlinear Schrödinger equation (NLSE). New expressions for the coefficients of the NLSE are derived and compared, for a Yukawa-type potential energy and for a “dressed” potential energy, taking into account interaction and geometric nonlinearities.

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The dust lattice wave in dusty plasma crystals is reexamined, taking into account the dependence of the dust grain charge on the grain potential. The Poisson equation for small grain potentials then assumes the form of the Schrödinger equation. The spatial distribution of the potential in the lattice includes the effect of the whole system of dust particles. Such a self-consistent description gives the dispersion relation for the dust lattice wave, which is different from the expression found earlier. The case of large grain charge is also considered. The frequency of the lattice oscillation increases considerably for large grain charges. Furthermore, it is noted that an ideal lattice can only exist if the dusty plasma parameters satisfy a definite relation between the dusty plasma Debye radius, the intergrain separation, and the grain size.