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Linear and nonlinear studies of dust lattice waves in a dusty plasma crystal have been carried out on the basis of the Schr?dinger equation which is deduced from Poisson's equation for small dust grain potentials. The spatial distribution of the potential in the dust-lattice includes the effect of the whole system of the dust particles. Such a self-consistent analysis gives a dispersion relation for the dust lattice wave, which is different from the expression found earlier. The frequency of the lattice oscillation increases considerably for large grain charges. Furthermore, it is found that an ideal lattice can only exist if the dusty plasma parameters satisfy a definite relationship between the dusty plasma Debye radius, the inter-grain separation, and the grain size. Finally, accounting for the weak nonlinearities, we also derive a Korteweg-de Vries (KdV) equation for the nonlinear dust lattice waves in the long wavelength approximation $(kv\ll1)$, where k is the wave number and d the inter-grain spacing.
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