Chaos, Solitons and Fractals 130 (2020) 109448 Contents lists available at ScienceDirect Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos Analytical solution for nonplanar waves in a plasma with q-nonextensive nonthermal velocity distribution:Weighted residual method Hilmi Demiray Isik University, Faculty of Arts and Sciences, Department of Mathematics, Sile 34980 Istanbul, Turkey a r t i c l e i n f o Article history: Received 8 March 2019 Revised 29 July 2019 Accepted 13 September 2019 Keywords: Nonplanar solitary waves Cairns-Tsallis distribution q-nonextensive nonthermal distribution a b s t r a c t The basic nonlinear equations describing the dynamics of a two component plasma consisting of cold positive ions and electrons obeying hybrid q- nonextensive nonthermal velocity distribution are examined in the cylindrical(spherical) coordinates through the use of reductive perturbation method and the cylin- drical(spherical) KdV and the modified KdV equations are obtained. An approximate analytical method for the progressive wave solution is presented for these evolution equation in the sense of weighted residual method. It is observed that both amplitudes and the wave speeds decrease with the time parameter τ . Since the wave profiles change with τ , the waves cannot be treated as solitons. It is further observed that the amplitudes of spherical waves are larger than those of the cylindrical waves; and the wave ampli- tudes of modified KdV equation are much larger than those of the KdV equation. The effects of physical parameters (α, q) on the wave characteristics are also discussed. © 2019 Elsevier Ltd. All rights reserved. 1. Introduction Solitons are localized pulse shaped stable nonlinear entities which arise as a manifestation of balance between the nonlin- earity and dispersion. Washimi and Taniuti [1] were the first to use the reductive perturbation method to derive the Korteweg- deVries(KdV) equation for ion-acoustic solitons(IASs) in plasma. Early investigations on IASs were based on the Maxwellian distri- bution, which are believed to be universally valid. However, the recent studies on space and laboratory plasmas indicate the pres- ence of energetic particles in tailed- particle distribution. Moreover, the observations made by Viking spacecraft [2] and Freja satel- lite [3] showed the importance of electrostatic solitary structures. Cairns et al. [4] introduced a distribution model in terms of a pa- rameter α, which measures the deviation from the Maxwellian dis- tribution function. In another development, Tribeche et al. [5] ex- tended the work of Cairns et al. [4] and introduced a hybrid Cairns- Tsallis distribution. Using this model Wang et al. [6], Saha et al. [7], Saha and Tamang [8], and Tamang et al. [9] showed the existence of electron-acoustic solitary waves in such a plasma. It was further shown that the regions of existence and the amplitude of the soli- tary waves are effected by the nonextensive parameter q and the nonthermal parameter α. More recently Bouzit et al. [10,11] have E-mail address: hilmi.demiray@isikun.edu.tr investigated the effect of an interplay between nonthermality and nonextensivity on ion-acoustic solitons. They further observed that, ion-acoustic solitons exhibit compression or rarefaction, depending on the nonextensivity and nonthermality of modulational instabil- ity of ion acoustic waves. Moreover, it is shown that both nonther- mal and nonextensive parameters affect the domains of instabil- ities. In all these works the planar field equations of plasmas in one dimension are investigated. The propagation of nonlinear waves in cylindrically and spher- ically symmetric plasmas had been studied before by several re- searchers (see, for instance, Maxon and Viecelli [12,13], Mamun and Shukla [14], Sahu and Roychoudhury [15] and Xue [16]) and obtained the cylindrical and the spherical KdV equations. Presently, there is no analytical solutions available for these evolution equa- tions but some numerical solutions exist. Demiray and Bayindir [17] and Demiray and El-Zahar [18] presented some approximate analytical solutions based on the weighted residual methods and the results are compared with the results of numerical solutions. It is observed that the analytical results agree well with the nu- merical solutions. In the present work, employing the basic nonlinear equations describing the dynamics of a two component plasma consisting of cold positive ions and electrons obeying hybrid q- nonexten- sive nonthermal velocity distribution are examined in the cylindri- cal(spherical) coordinates through the use of reductive perturba- tion method and the cylindrical(spherical) KdV and modified KdV https://doi.org/10.1016/j.chaos.2019.109448 0960-0779/© 2019 Elsevier Ltd. All rights reserved.