{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T23:16:54Z","timestamp":1780442214896,"version":"3.54.1"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,5,28]],"date-time":"2021-05-28T00:00:00Z","timestamp":1622160000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The soliton waves\u2019 physical behavior on the pseudo spherical surfaces is studied through the analytical solutions of the nonlinear (1+1)\u2013dimensional Kaup\u2013Kupershmidt (KK) equation. This model is named after Boris Abram Kupershmidt and David J. Kaup. This model has been used in various branches such as fluid dynamics, nonlinear optics, and plasma physics. The model\u2019s computational solutions are obtained by employing two recent analytical methods. Additionally, the solutions\u2019 accuracy is checked by comparing the analytical and approximate solutions. The soliton waves\u2019 characterizations are illustrated by some sketches such as polar, spherical, contour, two, and three-dimensional plots. The paper\u2019s novelty is shown by comparing our obtained solutions with those previously published of the considered model.<\/jats:p>","DOI":"10.3390\/sym13060963","type":"journal-article","created":{"date-parts":[[2021,5,31]],"date-time":"2021-05-31T03:45:29Z","timestamp":1622432729000},"page":"963","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["Bright\u2013Dark Soliton Waves\u2019 Dynamics in Pseudo Spherical Surfaces through the Nonlinear Kaup\u2013Kupershmidt Equation"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8466-168X","authenticated-orcid":false,"given":"Mostafa M. A.","family":"Khater","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Jiangsu University, Zhenjiang 212013, China"},{"name":"Department of Mathematics, Obour Institutes, Cairo 11828, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5920-250X","authenticated-orcid":false,"given":"Lanre","family":"Akinyemi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Prairie View A & M University, Prairie View, TX 77446, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4781-5445","authenticated-orcid":false,"given":"Sayed K.","family":"Elagan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8115-0638","authenticated-orcid":false,"given":"Mohammed A.","family":"El-Shorbagy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia"},{"name":"Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom 32511, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Suleman H.","family":"Alfalqi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, Mahayil Asir King Khalid University, Abha 61421, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jameel F.","family":"Alzaidi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, Mahayil Asir King Khalid University, Abha 61421, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8373-1428","authenticated-orcid":false,"given":"Nawal A.","family":"Alshehri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/s13360-020-00891-x","article-title":"A new fractional nonlinear singular heat conduction model for the human head considering the effect of febrifuge","volume":"135","author":"Wang","year":"2020","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1425","DOI":"10.1016\/j.aej.2020.03.046","article-title":"Dynamical analysis of the nonlinear complex fractional emerging telecommunication model with higher\u2013order dispersive cubic\u2013quintic","volume":"59","author":"Park","year":"2020","journal-title":"Alex. Eng. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11082-019-2162-8","article-title":"Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique","volume":"52","author":"Gao","year":"2020","journal-title":"Opt. Quantum Electron."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1016\/j.ijleo.2018.05.100","article-title":"Comment on four papers of Elsayed ME Zayed, Abdul-Ghani Al-Nowehy, Reham MA Shohib and Khaled AE Alurrfi (Optik 130 (2017) 1295\u20131311 & Optik 143 (2017) 84\u2013103 & Optik 158 (2018) 970\u2013984 & optik 144 (2017) 132\u2013148)","volume":"172","author":"Khater","year":"2018","journal-title":"Optik"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Khater, M.M., Hamed, Y.S., and Lu, D. (2020). On rigorous computational and numerical solutions for the voltages of the electrified transmission range with the day yet distance. Numer. Methods Partial. Differ. Equ., 1\u201310.","DOI":"10.1002\/num.22700"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"103954","DOI":"10.1016\/j.rinp.2021.103954","article-title":"Analytical and semi-analytical solutions for Phi-four equation through three recent schemes","volume":"22","author":"Khater","year":"2021","journal-title":"Results Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"103890","DOI":"10.1016\/j.rinp.2021.103890","article-title":"Abundant stable computational solutions of Atangana\u2013Baleanu fractional nonlinear HIV-1 infection of CD4+ T\u2013cells of immunodeficiency syndrome","volume":"22","author":"Khater","year":"2021","journal-title":"Results Phys."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2150220","DOI":"10.1142\/S0217984921502201","article-title":"Diverse solitary and Jacobian solutions in a continually laminated fluid with respect to shear flows through the Ostrovsky equation","volume":"35","author":"Khater","year":"2021","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"025130","DOI":"10.1063\/5.0038671","article-title":"Some optical soliton solutions to the perturbed nonlinear Schr\u00f6dinger equation by modified Khater method","volume":"11","author":"Khater","year":"2021","journal-title":"AIP Adv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s12596-020-00667-7","article-title":"Optical soliton structure of the sub-10-fs-pulse propagation model","volume":"50","author":"Khater","year":"2021","journal-title":"J. Opt."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"015223","DOI":"10.1063\/5.0036261","article-title":"Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model","volume":"11","author":"Chu","year":"2021","journal-title":"AIP Adv."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Attia, R.A., Baleanu, D., Lu, D., Khater, M.M., and Ahmed, E.S. (2021). Computational and numerical simulations for the deoxyribonucleic acid (DNA) model. Discret. Contin. Dyn. Syst. S.","DOI":"10.3934\/dcdss.2021018"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2682","DOI":"10.1002\/mma.6951","article-title":"Analytical and semi-analytical solutions for time-fractional Cahn\u2013Allen equation","volume":"44","author":"Khater","year":"2021","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"103317","DOI":"10.1016\/j.rinp.2020.103317","article-title":"On the numerical investigation of the interaction in plasma between (high & low) frequency of (Langmuir & ion-acoustic) waves","volume":"18","author":"Khater","year":"2020","journal-title":"Results Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-020-02705-x","article-title":"On abundant new solutions of two fractional complex models","volume":"2020","author":"Khater","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"055213","DOI":"10.1088\/1402-4896\/ab76f8","article-title":"Computational and numerical simulations for the nonlinear fractional Kolmogorov\u2013Petrovskii\u2013Piskunov (FKPP) equation","volume":"95","author":"Khater","year":"2020","journal-title":"Phys. Scr."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2050123","DOI":"10.1142\/S0217984920501237","article-title":"The new structure of analytical and semi-analytical solutions of the longitudinal plasma wave equation in a magneto-electro-elastic circular rod","volume":"34","author":"Li","year":"2020","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"2050096","DOI":"10.1142\/S0217979220500964","article-title":"Novel soliton waves of two fluid nonlinear evolutions models in the view of computational scheme","volume":"34","author":"Khater","year":"2020","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_19","first-page":"1","article-title":"Abundant distinct types of solutions for the nervous biological fractional FitzHugh\u2013Nagumo equation via three different sorts of schemes","volume":"2020","author":"Khater","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"045216","DOI":"10.1063\/1.5142796","article-title":"Computational simulations of the couple Boiti\u2013Leon\u2013Pempinelli (BLP) system and the (3+1)-dimensional Kadomtsev\u2013Petviashvili (KP) equation","volume":"10","author":"Yue","year":"2020","journal-title":"AIP Adv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2050084","DOI":"10.1142\/S0217979220500848","article-title":"Abundant analytical solutions of the fractional nonlinear (2+1)-dimensional BLMP equation arising in incompressible fluid","volume":"34","author":"Yue","year":"2020","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"105120","DOI":"10.1063\/5.0024417","article-title":"Two effective computational schemes for a prototype of an excitable system","volume":"10","author":"Khater","year":"2020","journal-title":"AIP Adv."},{"key":"ref_23","first-page":"1","article-title":"Oblique explicit wave solutions of the fractional biological population (BP) and equal width (EW) models","volume":"2020","author":"Khater","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"110676","DOI":"10.1016\/j.chaos.2021.110676","article-title":"On semi analytical and numerical simulations for a mathematical biological model; the time-fractional nonlinear Kolmogorov\u2013Petrovskii\u2013Piskunov (KPP) equation","volume":"144","author":"Khater","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Khater, M.M., Inc, M., Nisar, K., and Attia, R.A. (2021). Multi\u2013solitons, lumps, and breath solutions of the water wave propagation with surface tension via four recent computational schemes. Ain Shams Eng. J.","DOI":"10.1016\/j.asej.2020.10.029"},{"key":"ref_26","unstructured":"Mohyud-Din, S.T., Y\u0131ld\u0131r\u0131m, A., and Sar\u0131ayd\u0131n, S. (2011). Numerical soliton solution of the Kaup-Kupershmidt equation. Int. J. Numer. Methods Heat Fluid Flow."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/s10255-018-0757-7","article-title":"Conservatory of Kaup-Kupershmidt equation to the concept of fractional derivative with and without singular kernel","volume":"34","author":"Atangana","year":"2018","journal-title":"Acta Math. Appl. Sin. Engl. Ser."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"073507","DOI":"10.1063\/1.1939988","article-title":"Nonlocal symmetries and the Kaup\u2013Kupershmidt equation","volume":"46","author":"Reyes","year":"2005","journal-title":"J. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1002\/num.22580","article-title":"An efficient computational technique for time-fractional Kaup-Kupershmidt equation","volume":"37","author":"Prakasha","year":"2021","journal-title":"Numer. Methods Partial Differ. Equ."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"102372","DOI":"10.1016\/j.rinp.2019.102372","article-title":"Applications of dispersive analytical wave solutions of nonlinear seventh order Lax and Kaup-Kupershmidt dynamical wave equations","volume":"14","author":"Aljahdaly","year":"2019","journal-title":"Results Phys."},{"key":"ref_31","first-page":"72","article-title":"On numerical soliton solution of the Kaup\u2013Kupershmidt equation and convergence analysis of the decomposition method","volume":"172","year":"2006","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0167-2789(00)00081-6","article-title":"Nonlinear superposition formula for the Kaup\u2013Kupershmidt partial differential equation","volume":"144","author":"Musette","year":"2000","journal-title":"Phys. D Nonlinear Phenom."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/963\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:09:52Z","timestamp":1760162992000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/963"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,28]]},"references-count":32,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13060963"],"URL":"https:\/\/doi.org\/10.3390\/sym13060963","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,5,28]]}}}