Abstract The nonlinear dynamical exact wave solutions to the non-fractional order and the time-fractional order of the biological population models are achieved for the first time in the framwork of the Paul-Painlevé approach method (PPAM). When the variables appearing in the exact solutions take specific values, the solitary wave solutions will be easily obtained. The realized results prove the efficiency of this technique.
Corresponding Authors:
Ahmet Bekir
E-mail: bekirahmet@gmail.com
Cite this article:
Ahmet Bekir and Emad H M Zahran Nonlinear dynamical wave structures of Zoomeron equation for population models 2022 Chin. Phys. B 31 060401
[1] Calogero F and Degasperis A 1976 Nuovo Cimento B32 201 [2] Shakeri E and Dehghan M 2007 Comput. Math. Appl.54 1197 [3] Bekir A and Guner O 2013 Chin. Phys. B22 110202 [4] Jabbari A, Kheiri H and Bekir A 2011 Comput. Math. Appl.62 2177 [5] Bekir A and Zahran E H M 2020 Optik223 165233 [6] Li N and Liu X Q 2013 Acta Phys. Sin.62 160203 (in Chinese) [7] Bekir A and Zahran E H M 2021 Optik240 166939 [8] Song C Q, Xiao D M and Zhu Z N 2017 Chin. Phys. B26 100204 [9] Bekir A and Zahran E H M 2021 Phys. Scr.96 055212 [10] Song M, Beidan W and Jun C 2020 Chin. Phys. B29 100206 [11] Shehata M S M, Rezazadeh H, Zahran E H M, Tala-Tebue E and Bekir A 2019 Commun. Theor. Phys.71 1275 [12] Bekir A and Zahran E H M 2021 Pramana J. Phys.95 158 [13] Shehata M S M, Rezazadeh H, Jawad A J M, Zahran E H M and Bekir A 2021 Revista Mexicana de Fisica67 050704 [14] Huang L L, Qiao Z J and Chen Y 2018 Chin. Phys. B27 020201 [15] Zahran E H M and Khater M M 2016 Appl. Math. Model.40 1769 [16] Khater M M, Lu D and Zahran E H M 2017 Appl. Math. Inf. Sci.11 1347 [17] Zahran E H M and Bekir A 2021 Opt. Quantum Electron.53 680 [18] Wazwaz A M 2004 Appl. Math. Comput.154 713 [19] Bekir A, Zahran E H M and Güner O 2020 Int. J. Mod. Phys. B35 2150025 [20] Zhang S, Jing-Lin T and Wei W 2008 Phys. Lett. A372 2254 [21] Fan E 2000 Phys. Lett. A277 212 [22] Abazari R 2011 Appl. Math. Sci.5 2943 [23] Khan K, Akbar M A, Salam A and Islam H 2014 Ain Shams Engineering Journal5 877 [24] Bi Y, Zhang Z, Liu Q and Liu T 2021 Commun. Nonlinear Sci. Numer. Simul.102 105918 [25] Wenhui Z, Hongwei Y and Dezhi G 2020 Mathematical Modeling and Its Applications466 73 [26] Bekir A, Ayhan B and Ozer M N 2013 Chin. Phys. B22 010202 [27] Bekir A and Zahran E H M 2020 Int. J. Mod. Phys. B34 2050270 [28] Bekir A and Zahran E H M 2021 Opt. Quantum Electron.53 118 [29] Caputo M and Mainardi F 1971 Revista del Nuovo Cimento161 198 [30] Caputo M 1967 J. Royal Astronom. Soc.13529 539 [31] Samko S G, Kilbas A A and Marichev O I 1993 Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach Science Publishers, Switzerland) [32] Jumarie G 2006 Comput. Math. Appl. 51 1367 [33] Jumarie G 2006 Comput. Math. Appl.51 1367 [34] Jumarie G 2009 Appl. Math. Lett.22 378 [35] Kilbas A A, Srivastava H M and Trujillo J J 2006 Theory and Applications of Fractional Differential Equations-North-Holland Mathematical Studies 204 (Amsterdam: Elsevier) [36] Bekir A and Guner O 2014 Ain Shams Eng. Journal5 959 [37] Gupta V G and Kumar P 2013 International Journal of Science and Research6 14 [38] Gurtin M Eand Maccamy R C 1977 Math. Biol. Sci.54 35 [39] Lu Y G 2000 Appl. Math. Lett.13 123 [40] Liu Y, Li Z and Zhang Y 2011 Fract. Diff. Calc.1 117 [41] Khalil R, Al Horani M, Yousef A and Sababheh M 2014 J. Comput. Appl. Math.265 65 [42] Ain Q T and He J H 2019 Thermal Science23 1707
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
