We present a theoretical study of the generation of ion Bernstein waves (IBWs) and ion cyclotron waves (ICWs) through the parametric instability of whistler waves in a dusty plasma. The interaction begins with electron density perturbations associated with the IBW and the oscillatory velocity of plasma electrons driven by the whistler pump wave. This combination generates a non-linear current that excites the whistler sideband wave. The whistler pump wave and its sideband exert a ponderomotive force on the electrons, generating ion Bernstein and ICWs. The influence of dust charge fluctuations on this wave generation process is systematically investigated. Growth rate expressions for ion Bernstein and ICWs are derived, revealing key dependencies. The growth rate increases with higher pump wave amplitude, wave number, scattering angle, and dust grain density. In contrast, an increase in the size of the dust grain and the temperature of the electrons reduce the growth rate. A comparative analysis shows that the IBW exhibits greater instability than the ICW, resulting in a higher growth rate for the IBW in dusty plasma. These findings provide deeper insights into wave dynamics in dusty plasma environments, with potential implications for space and laboratory plasma studies.
Classification of sideband instability regimes for whistler waves with trapped electrons