Investigation of Nonlinear Dust-Acoustic Waves in the Presence of an Electron Beam

The nonlinear effects of an unmagnetized dusty plasma including heavy dust grains, electrons, ions in the existence of an electron beam are examined. The Korteweg–de Vries (KdV) equation governing this system isobtained utilizing thereductive perturbation technique (RPT). A planar dynamical system (DS) is established for the KdV equation employing a traveling wave transformation. Phase portrait is established and the associated nonlinear waves are investigated. The impacts of dust density ($N_{0d}$), cathode potential ($\varphi _{c}$), and phase velocity ($V$) on nonlinear wave features are shown. It is observed that increasing value of$N_{0d}$enhances amplitude of dust-acoustic solitary wave (DASW) and nonlinear periodic dust-acoustic wave (NPDAW), with a constant width of DASW and a higher width of NPDAW. Also, higher values of$\varphi _{c}$reduce the amplitude of both waves, with no change in width of DASW and a reduced width of NPDAW. Increasing value of$V$results in larger amplitude and width for both DASW and NPDAW. Two-soliton solutions, obtained via the Hirota direct method, show that for small values of time ($\tau $), the solitons are separated, merge at$\tau = 0$, and then diverge as$\tau $increases. The phase shift decreases as$\theta $increases forvariousvalues of electron ratio [$\alpha = N_{0e}/(Z_{d} N_{0d})$] and ion ratio [$\beta = N_{0i}/(Z_{d} N_{0d})$]. Our results show that the electron beam significantly affects amplitude of the wave, phase speed, and soliton propagation. Adjusting cathode voltage and electron beam density can control the transmission of acoustic waves in dusty plasmas. The achieved results can be used in plasma laser interaction.