The dust acoustic waves with (r, q) distributed ions and Maxwellian electrons are studied for dusty plasmas found in planetary rings. The expression of polarization force in a dusty plasma, modified due to (r, q) distributed ions, is evaluated for the first time. The fundamental model equations are reduced to a modified Korteweg–de Vries (mKdV) equation by applying the familiar reductive perturbation technique. The dynamical system analysis provides phase portraits, effective potential, compressive and rarefactive solitary waves, and nonlinear periodic waves for the dusty plasma. Hirota's bilinear formalism is utilized to generate the multi-soliton solutions. The parametric analysis of dust acoustic solitary waves (DASWs) shows that the modified polarization force tends to augment the amplitude of the solitary waves. The comparison of kappa, (r, q) and Maxwellian distributions shows that the DASWs with kappa distribution exhibit minimum amplitude, while DASWs with higher values of indices in (r, q) ion distribution manifest maximum amplitude of solitary waves. The Hirota breather is formed using the complex-conjugate pair of propagation vectors for the two-soliton solution of the mKdV equation. The interaction between solitons, between a breather and a soliton, and between two breathers shows that such interactions have nonlinear superposition characteristics. This shows that the breather solutions are stable solutions, like solitons, for an integrable nonlinear partial differential equation (nPDE). The model is applied to the dusty space plasmas; however, the soliton and breather solutions obtained in this paper are general and may be applied to the systems where integrable nPDEs are obtained.
Jahangir et al. (Fri,) studied this question.