{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:16:57Z","timestamp":1774311417271,"version":"3.50.1"},"reference-count":46,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,8,1]],"date-time":"2026-08-01T00:00:00Z","timestamp":1785542400000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of Computational and Applied Mathematics"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1016\/j.cam.2025.117250","type":"journal-article","created":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T17:08:01Z","timestamp":1766164081000},"page":"117250","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"special_numbering":"C","title":["Auto-B\u00e4cklund transformation, kink, anti-kink and periodic wave solutions for Painlev\u00e9 integrable\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" altimg=\"si1.svg\">\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mn>3<\/mml:mn>\n                        <mml:mo linebreak=\"goodbreak\">+<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    -dimensional variable coefficients combined pKP-BKP equations in nonlinear waves"],"prefix":"10.1016","volume":"481","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4350-604X","authenticated-orcid":false,"given":"Shailendra","family":"Singh","sequence":"first","affiliation":[]},{"given":"Abdul-Majid","family":"Wazwaz","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.cam.2025.117250_bib0001","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2024.109232","article-title":"Painlev\u00e9 analysis of the resonant third-order nonlinear Schr\u00f6dinger equation","volume":"158","author":"Kudryashov","year":"2024","journal-title":"Appl. Math. Lett."},{"issue":"5","key":"10.1016\/j.cam.2025.117250_bib0002","doi-asserted-by":"crossref","first-page":"2177","DOI":"10.1108\/HFF-01-2024-0053","article-title":"Extended (3+1)-dimensional Kairat-II and Kairat-X equations: Painlev\u00e9 integrability, multiple soliton solutions, lump solutions, and breather wave solutions","volume":"34","author":"Wazwaz","year":"2024","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"key":"10.1016\/j.cam.2025.117250_bib0003","doi-asserted-by":"crossref","DOI":"10.1016\/j.rinp.2024.107640","article-title":"New analytical technique for prototype closed form solutions of certain nonlinear partial differential equations","volume":"60","author":"Mumtaz","year":"2024","journal-title":"Results Phys."},{"key":"10.1016\/j.cam.2025.117250_bib0004","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2025.109615","article-title":"Inhomogeneity, magnetic auto-B\u00e4cklund transformations and magnetic solitons for a generalized variable-coefficient Kraenkel-Manna-Merle system in a deformed ferrite","volume":"171","author":"Gao","year":"2025","journal-title":"Appl. Math. Lett."},{"issue":"1","key":"10.1016\/j.cam.2025.117250_bib0005","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/s13344-025-0012-y","article-title":"In an ocean or a river: Bilinear auto-B\u00e4cklund transformations and similarity reductions on an extended time-dependent (3+1)-dimensional shallow water wave equation","volume":"39","author":"Gao","year":"2025","journal-title":"China Ocean Eng."},{"key":"10.1016\/j.cam.2025.117250_bib0006","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1016\/j.cjph.2024.10.004","article-title":"Hetero-B\u00e4cklund transformation, bilinear forms and multi-solitons for a (2+1)-dimensional generalized modified dispersive water-wave system for the shallow water","volume":"92","author":"Gao","year":"2024","journal-title":"Chin. J. Phys."},{"key":"10.1016\/j.cam.2025.117250_bib0007","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s13344-025-0057-y","article-title":"Open-ocean shallow-water dynamics via a (2+1)-dimensional generalized variable-coefficient Hirota-Satsuma-Ito system: oceanic auto-B\u00e4cklund transformation and oceanic solitons","volume":"39","author":"Gao","year":"2025","journal-title":"China Ocean Eng."},{"issue":"5","key":"10.1016\/j.cam.2025.117250_bib0008","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1007\/s12346-025-01385-w","article-title":"Bilinear forms, bilinear auto-B\u00e4cklund transformations and similarity reductions for a (3+1)-dimensional generalized variable-coefficient Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in quantum and fluid mechanics","volume":"24","author":"Gao","year":"2025","journal-title":"Qual. Theory Dyn. Syst."},{"key":"10.1016\/j.cam.2025.117250_bib0009","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2024.109018","article-title":"Two-layer-liquid and lattice considerations through a (3+1)-dimensional generalized Yu-Toda-Sasa-Fukuyama system","volume":"152","author":"Gao","year":"2024","journal-title":"Appl. Math. Lett."},{"key":"10.1016\/j.cam.2025.117250_bib0010","doi-asserted-by":"crossref","DOI":"10.1016\/j.aml.2025.109720","article-title":"A new (2+1)-dimensional like-Harry-Dym equation with derivation and soliton solutions","volume":"172","author":"Wang","year":"2026","journal-title":"Appl. Math. Lett."},{"issue":"2","key":"10.1016\/j.cam.2025.117250_bib0011","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/s12346-025-01241-x","article-title":"Bilinear B\u00e4cklund transformations, as well as N-soliton, breather, fission\/fusion and hybrid solutions for a (3+1)-dimensional integrable wave equation in a fluid","volume":"24","author":"Feng","year":"2025","journal-title":"Qual. Theory Dyn. Syst."},{"key":"10.1016\/j.cam.2025.117250_bib0012","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1007\/s12346-024-01125-6","article-title":"N-soliton and other analytic solutions for a (3+1)-dimensional Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation with the time-dependent coefficients for the shallow water waves","volume":"23","author":"Shan","year":"2024","journal-title":"Qual. Theory Dyn. Syst."},{"issue":"4","key":"10.1016\/j.cam.2025.117250_bib0013","doi-asserted-by":"crossref","first-page":"3655","DOI":"10.1007\/s11071-024-10397-1","article-title":"N-soliton, hth-order breather, hybrid and multi-pole solutions for a generalized variable-coefficient Gardner equation with an external force in a plasma or fluid","volume":"113","author":"Liu","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"10.1016\/j.cam.2025.117250_bib0014","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.114745","article-title":"The lie point symmetry criteria and formation of exact analytical solutions for Kairat-II equation: Paul-Painlev\u00e9 approach","volume":"182","author":"Faridi","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"10.1016\/j.cam.2025.117250_bib0015","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1016\/j.cjph.2024.09.004","article-title":"Generalization and analytic exploration of soliton solutions for nonlinear evolution equations via a novel symbolic approach in fluids and nonlinear sciences","volume":"92","author":"Mohan","year":"2024","journal-title":"Chin. J. Phys."},{"issue":"9","key":"10.1016\/j.cam.2025.117250_bib0016","doi-asserted-by":"crossref","DOI":"10.1088\/1402-4896\/ad6a9f","article-title":"Integrability and multiple kinks, lumps, and breathers solutions to an extended (3+1)-dimensional Calogero-Bogoyavlenskii-Schiff fluid model","volume":"99","author":"Wazwaz","year":"2024","journal-title":"Phys. Scr."},{"issue":"3","key":"10.1016\/j.cam.2025.117250_bib0017","article-title":"Bright and dark optical soliton solutions for a nonlinear Schr\u00f6dinger equation with Kerr law nonlinearity and weak nonlocality","volume":"38","author":"Wazwaz","year":"2025","journal-title":"Int. J. Numer. Modell. Electron. Netw. Devices Fields"},{"issue":"11","key":"10.1016\/j.cam.2025.117250_bib0018","doi-asserted-by":"crossref","DOI":"10.1016\/j.asej.2025.103618","article-title":"Similarity reduction, B\u00e4cklund transformation and solitary waves for a generalized shallow water wave equation with the variable coefficients","volume":"16","author":"El-Shiekh","year":"2025","journal-title":"Ain Shams Eng. J."},{"issue":"10","key":"10.1016\/j.cam.2025.117250_bib0019","doi-asserted-by":"crossref","first-page":"23869","DOI":"10.3934\/math.20251061","article-title":"Similarity reduction and novel Jacobi wave solutions for the variable (4+1)-dimensional Fokas equation","volume":"10","author":"El-Shiekh","year":"2025","journal-title":"AIMS Math."},{"issue":"14","key":"10.1016\/j.cam.2025.117250_bib0020","doi-asserted-by":"crossref","first-page":"11530","DOI":"10.1002\/mma.10141","article-title":"Novel nonlinear quantum dust acoustic waves for modified variable coefficients Zakharove-Kusnetsov equation in dusty plasma","volume":"47","author":"Gaballah","year":"2024","journal-title":"Math. Methods Appl. Sci."},{"issue":"5","key":"10.1016\/j.cam.2025.117250_bib0021","doi-asserted-by":"crossref","first-page":"872","DOI":"10.1007\/s11082-024-06789-7","article-title":"Bilinear form and N-soliton thermophoric waves for the variable coefficients (2+1)-dimensional graphene sheets equation","volume":"56","author":"El-Shiekh","year":"2024","journal-title":"Opt. Quantum Electron."},{"issue":"1","key":"10.1016\/j.cam.2025.117250_bib0022","doi-asserted-by":"crossref","DOI":"10.1080\/16583655.2023.2192280","article-title":"Similarity reduction and multiple novel travelling and solitary wave solutions for the two-dimensional Bogoyavlensky-Konopelchenko equation with variable coefficients","volume":"17","author":"Gaballah","year":"2023","journal-title":"J. Taibah Univ. Sci."},{"key":"10.1016\/j.cam.2025.117250_bib0023","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.aej.2024.08.052","article-title":"Painlev\u00e9 integrability and multiple soliton solutions for the extensions of the (modified) Korteweg-de Vries-type equations with second-order time-derivative","volume":"103","author":"Wazwaz","year":"2024","journal-title":"Alexandria Eng. J."},{"key":"10.1016\/j.cam.2025.117250_bib0024","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2025.116307","article-title":"Multiple soliton solutions and other scientific solutions for a new Painlev\u00e9 integrable fifth-order equation","volume":"196","author":"Wazwaz","year":"2025","journal-title":"Chaos Solitons Fractals"},{"issue":"7","key":"10.1016\/j.cam.2025.117250_bib0025","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/s10773-024-05670-3","article-title":"Painlev\u00e9 analysis, B\u00e4cklund transformation and soliton solutions of the (2+1)-dimensional variable-coefficient Boussinesq equation","volume":"63","author":"Zhang","year":"2024","journal-title":"Int. J. Theor. Phys."},{"issue":"11","key":"10.1016\/j.cam.2025.117250_bib0026","doi-asserted-by":"crossref","first-page":"9355","DOI":"10.1007\/s11071-024-09450-w","article-title":"Painlev\u00e9 analysis, auto-B\u00e4cklund transformations, bilinear form and analytic solutions on some nonzero backgrounds for a (2+1)-dimensional generalized nonlinear evolution system in fluid mechanics and plasma physics","volume":"112","author":"Zhou","year":"2024","journal-title":"Nonlinear Dyn."},{"issue":"06","key":"10.1016\/j.cam.2025.117250_bib0027","doi-asserted-by":"crossref","DOI":"10.1142\/S0219887822500864","article-title":"New abundant analytic solutions for generalized KdV6 equation with time-dependent variable coefficients using Painlev\u00e9 analysis and auto-B\u00e4cklund transformation","volume":"19","author":"Singh","year":"2022","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"issue":"8","key":"10.1016\/j.cam.2025.117250_bib0028","doi-asserted-by":"crossref","DOI":"10.1088\/1402-4896\/ace6dc","article-title":"New analytical solutions and integrability for the (2+1)-dimensional variable coefficients generalized Nizhnik-Novikov-Veselov system arising in the study of fluid dynamics via auto-B\u00e4cklund transformation approach","volume":"98","author":"Singh","year":"2023","journal-title":"Phys. Scr."},{"issue":"2","key":"10.1016\/j.cam.2025.117250_bib0029","doi-asserted-by":"crossref","DOI":"10.1016\/j.jmaa.2023.127695","article-title":"Bilinear representation, bilinear B\u00e4cklund transformation, Lax pair and analytical solutions for the fourth-order potential Ito equation describing water waves via bell polynomials","volume":"530","author":"Singh","year":"2024","journal-title":"J. Math. Anal. Appl."},{"key":"10.1016\/j.cam.2025.117250_bib0030","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.114603","article-title":"The exact solutions for the nonlocal Kundu-NLS equation by the inverse scattering transform","volume":"180","author":"Li","year":"2024","journal-title":"Chaos Solitons Fractals"},{"issue":"2","key":"10.1016\/j.cam.2025.117250_bib0031","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1140\/epjp\/s13360-024-04953-2","article-title":"Superposition behavior of the lump solutions and multiple mixed function solutions for the (3+1)-dimensional Sharma-Tasso-Olver-like equation","volume":"139","author":"Han","year":"2024","journal-title":"Eur. Phys. J. Plus"},{"issue":"4","key":"10.1016\/j.cam.2025.117250_bib0032","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s12346-023-00823-x","article-title":"New analytic solutions for fluid flow equations in higher dimensions around an offshore structure describing bidirectional wave surfaces","volume":"22","author":"Singh","year":"2023","journal-title":"Qual. Theory Dyn. Syst."},{"issue":"10","key":"10.1016\/j.cam.2025.117250_bib0033","doi-asserted-by":"crossref","first-page":"11999","DOI":"10.1007\/s11071-024-10697-6","article-title":"Analysis of soliton wave structure for coupled higgs equation via lie symmetry, Paul Painlev\u00e9 approach and the unified method","volume":"113","author":"Shagolshem","year":"2025","journal-title":"Nonlinear Dyn."},{"issue":"5","key":"10.1016\/j.cam.2025.117250_bib0034","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1007\/s12346-025-01383-y","article-title":"Two distinct approaches for analytical solutions and Painlev\u00e9 integrability of the (3+1)-dimensional variable coefficient KP equation describing long waves in shallow water","volume":"24","author":"Singh","year":"2025","journal-title":"Qual. Theory Dyn. Syst."},{"issue":"10","key":"10.1016\/j.cam.2025.117250_bib0035","doi-asserted-by":"crossref","first-page":"11893","DOI":"10.1007\/s11071-024-10645-4","article-title":"Painlev\u00e9 analysis, restricted bright-dark N-solitons, and N-rogue waves of a (4+1)-dimensional variable-coefficient generalized KP equation in nonlinear sciences","volume":"113","author":"Mohan","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"10.1016\/j.cam.2025.117250_bib0036","doi-asserted-by":"crossref","DOI":"10.1016\/j.geomphys.2021.104191","article-title":"N-soliton solution of a combined pKP-BKP equation","volume":"165","author":"Ma","year":"2021","journal-title":"J. Geom. Phys."},{"key":"10.1016\/j.cam.2025.117250_bib0037","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2024.114886","article-title":"Breather wave solutions for an integrable (3+1)-dimensional combined pKP-BKP equation","volume":"182","author":"Wazwaz","year":"2024","journal-title":"Chaos Solitons Fractals"},{"issue":"12","key":"10.1016\/j.cam.2025.117250_bib0038","doi-asserted-by":"crossref","DOI":"10.1088\/0256-307X\/40\/12\/120501","article-title":"New Painlev\u00e9 integrable (3 1)-dimensional combined pKP-BKP equation: lump and multiple soliton solutions","volume":"40","author":"Wazwaz","year":"2023","journal-title":"Chin. Phys. Lett."},{"issue":"3","key":"10.1016\/j.cam.2025.117250_bib0039","doi-asserted-by":"crossref","DOI":"10.1088\/1674-1056\/ada43b","article-title":"Abundant invariant solutions of (3+1)-dimensional combined pKP-BKP equation","volume":"34","author":"Hu","year":"2025","journal-title":"Chin. Phys. B"},{"issue":"5","key":"10.1016\/j.cam.2025.117250_bib0040","doi-asserted-by":"crossref","first-page":"1790","DOI":"10.1108\/HFF-03-2025-0148","article-title":"Exploring soliton solutions and dynamical interaction wave-form patterns of (3+1)-dimensional combined pKP-BKP equation","volume":"35","author":"Kaur","year":"2025","journal-title":"Int. J. Numer. Methods Heat Fluid Flow"},{"issue":"3","key":"10.1016\/j.cam.2025.117250_bib0041","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1063\/1.525721","article-title":"The Painlev\u00e9 property for partial differential equations","volume":"24","author":"Weiss","year":"1983","journal-title":"J. Math. Phys."},{"key":"10.1016\/j.cam.2025.117250_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2023.113947","article-title":"The Painlev\u00e9 integrability and abundant analytical solutions for the potential Kadomtsev-Petviashvili (pKP) type coupled system with variable coefficients arising in nonlinear physics","volume":"175","author":"Singh","year":"2023","journal-title":"Chaos Solitons Fractals"},{"issue":"6","key":"10.1016\/j.cam.2025.117250_bib0043","doi-asserted-by":"crossref","DOI":"10.1209\/0295-5075\/acaac2","article-title":"New abundant exact solutions for MCBS-nMCBS equation: Painlev\u00e9 analysis and auto-B\u00e4cklund transformation","volume":"140","author":"Singh","year":"2022","journal-title":"Europhys. Lett."},{"issue":"3","key":"10.1016\/j.cam.2025.117250_bib0044","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s12346-025-01313-y","article-title":"Extraction of lumps, solitons, and multi-peakons for an integrable shallow water wave equation","volume":"24","author":"Raza","year":"2025","journal-title":"Qual. Theory Dyn. Syst."},{"key":"10.1016\/j.cam.2025.117250_bib0045","doi-asserted-by":"crossref","DOI":"10.3389\/fams.2025.1568757","article-title":"Application of the Tanh-Coth and HSI methods in deriving exact analytical solutions for the STFKS equation","volume":"11","author":"Hassaballa","year":"2025","journal-title":"Front. Appl. Math. Stat."},{"issue":"08","key":"10.1016\/j.cam.2025.117250_bib0046","doi-asserted-by":"crossref","DOI":"10.1142\/S0217984924504487","article-title":"Traveling wave solutions of Fordy-Gibbons equation","volume":"39","author":"Cevikel","year":"2025","journal-title":"Mod. Phys. Lett. B"}],"container-title":["Journal of Computational and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042725007642?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0377042725007642?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T19:11:34Z","timestamp":1770059494000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0377042725007642"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8]]},"references-count":46,"alternative-id":["S0377042725007642"],"URL":"https:\/\/doi.org\/10.1016\/j.cam.2025.117250","relation":{},"ISSN":["0377-0427"],"issn-type":[{"value":"0377-0427","type":"print"}],"subject":[],"published":{"date-parts":[[2026,8]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Auto-B\u00e4cklund transformation, kink, anti-kink and periodic wave solutions for Painlev\u00e9 integrable -dimensional variable coefficients combined pKP-BKP equations in nonlinear waves","name":"articletitle","label":"Article Title"},{"value":"Journal of Computational and Applied Mathematics","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cam.2025.117250","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"117250"}}