The propagation of a nonlinear wave packet of dust lattice waves sDLWd in a two-dimensional hexagonal crystal is investigated. The dispersion relation and the group velocity for DLW are found for longitudinalm and transversen propagation directions. The reductive perturbation method is used to derive a s2+1d-dimensional nonlinear Schrodinger equation sNLSEd that governs the weak lynonlinear propagationof thewavepacket. This NLSE isused toinves tigatethe modula tional instability of the packet of DLW. It is found that the instability region is different for different propagation directions.

A nonlinear Schrodinger equation (NLSE) is derived for a nonlinear transmission line in which the nonlinear capac-itance C is a function of voltage. For a linear long wavelength perturbations to a solution of a NLSE, the instability region is given in this paper.

By using the potential method and the perturbation method under the condition of small amplitude and shallow water waves, we analytically get the KdV-type equation for a viscous shallow water. It indicates that for one soliton-like solution, its amplitude will decrease as it propagates away due to the viscous eeects of water.

As is well known, Korteweg-de Vries equation is a typical one which has planar solitary waves. By considering the higher-dimensional nonlinear waves, we studied a Kadomtsev-Petviashvili (KP) equation and found some interesting results which explain experimental results well enough. Two same amplitude soliton solution of KP equation explain resonance phenomena reported by some experiments. Two arbitrary amplitude soliton solution of KP equation is also obtained in this Letter, which canalsoresultsinresonance phenomena. The phase shift after interaction between two soliton are obtained theoretically in this Letter.It is in agreement with experimental results.

Areasonable normalization for a magnetized dusty plasma with many different dust grains is adopted, which varies self-consistently with the system parameters. A Zakharov-Kuznetsov equation for small but finite am-plitude dust acoustic waves is obtained for magnetized dusty plasma which contains different dust grains by using the reductive perturbation technique. We study the dust size distribution. Some comparisons are made between dusty plasma in which the dust size distribution is considered, and the monosized dusty plasma in which there is only one kind of dust grain whose size is the average dust size. This suggests that both soliton velocity and width are larger than that for monosized dusty plasma, but its amplitude is smaller than that for monosized dusty plasma. If there are positively charged dust grains, compressive solitary waves may exist. The velocity, amplitude, and width of a soliton in multidimensional form for a magnetized dusty plasma which contains many different dust grains are studied as well.