Nonlinear modulation of the langmuir waves in a dusty plasma

The nonlinear modulation of the Langmuir waves by the dust acoustic waves (DAWs) and the dust ion acoustic waves (DIAWs) has been investigated. The analytical solutions of envelope soliton have been found in the limit of small amplitude. For the coupling between the Langmuir waves and the DAWs, there are two regions of envelope soliton propagation. And there is only one region of propagation speed for the coupling with the DIAWs