## 1 Introduction  
During the last few decades, a notable improvement has been made regarding the dusty (complex) plasmas having negatively/positively charged dust grains, ion, electron, and neutral plasmas components. The examples of dusty plasma in astrophysics and space systems are asteroid regions, interstellar media, interplanetary spaces, dark molecular clouds, accretion disks, earth's surroundings, and industry plasmas [1; 2; 3; 4]. Inertia is produced by the dust particle mass, and the restoring force is provided by the pressure of electrons/ions in the dust-acoustic (DA) mode. [PERSON] et al. [5] studied a dusty plasma (unmagnetized) and theoretically predicted the low-phase-velocity dust acoustic (DA) waves (in comparison to both electron as well as ion thermal velocities). The laboratory experiments [6] demonstrated some nonlinear features of dust acoustic waves. Dust grains are embedded in a surrounding plasma and radiative background under typical experimental settings. Because of the charge they carry, they interact with electrons and ions. Photoelectric emission, due to collision with electrons/ions, and secondary emission are some of the competing processes that charge dust grains, depending on local conditions [6]. Massive dust particles can accumulate a huge number of plasma electrons with high mobility, resulting in a dust particle with a negative charge. Consequently, the positive ion current increases and the electron current decreases as a result of the negative dust-surface potential. The steady-state of the electron current \((I_{e})\) and ion current \((I_{l})\) is \(|I_{e}|=|I_{l}|\), which establishes the equilibrium dust charge. The bombardment of electrons and positive ions was postulated to be the cause of dust charge in 1994 [7], when DSP was theoretically and experimentally investigated. These studies have been further expanded [8; 9] to emphasize the importance of negatively charged ions in Maxwellian/non-Maxwellian-distributed plasmas. There are several plasma events in space, such as the interstellar medium, ionosphere, thermosphere, planetary magnetosphere, ionosphere, and magnetosheaths, and the non-Maxwellian velocity distribution is frequently observed [10; 11; 12]. A novel distribution, developed to describe super-thermal particles detected by the Viking [13] and Freja-satellite [14], has been presented by [PERSON] et al. [15]. It has been expressed as a parameter alpha (\(\alpha\)) that calculates the variance from the Maxwellian to non-Maxwellian. As a result, it has been observed that in the presence of non-thermal electrons, the characteristics of ion-sound-solitary waves can alter, resulting in nonlinear solitary waves with either an augmented or depleted density. The [PERSON] model has since been used in a number of studies [16; 17; 18; 19]. [PERSON] and [PERSON], for example, studied large-amplitude dust-acoustic (DA) solitary structures in negatively dusty plasmas, in either non-thermal ions or electrons [20]. Later, [21] further looked into large ion-acoustic (IA) solitary structures and double layers (DL) that have positive ions as well non-thermal electrons plasmas. [PERSON] and colleagues have also looked at the combined effect of positively dust grains and non-thermal electrons in dust-IA solitary waves and the presence of non-thermal ions in DA waves [16]. In recent years, statistically non-extensive events using the Boltzmann-Gibbs-Shannon (BGS) entropic dimension deviation have received much attention. In statistical mechanics, large thermodynamical quantities are proportional to the mass of matter present, e.g., the particle number \((N)\), the volume \((V)\), the magnetization \((M)\), the internal energy \((E)\), and so on. The temperature \((T)\), pressure \((P)\), and chemical potential \((\mu)\) are non-extensive quantities that are not dependent on the mass of matter.