{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T03:40:34Z","timestamp":1776397234794,"version":"3.51.2"},"reference-count":68,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,8,16]],"date-time":"2025-08-16T00:00:00Z","timestamp":1755302400000},"content-version":"vor","delay-in-days":227,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>This study presents the application of a novel Softmax method to obtain exact analytical solutions of the breaking soliton equation, a nonlinear partial differential equation that models complex wave phenomena. By transforming the governing equation into an ordinary differential equation using a traveling wave transformation and constructing a solution framework based on the softmax function, the method yields exact solutions in hyperbolic, trigonometric, exponential, rational, and polynomial forms. The derived solutions include diverse wave structures such as kink\u2010lump solitons, rogue waves, and affine polynomial profiles, showcasing the method\u2019s flexibility and capacity to capture complex nonlinear dynamics. The softmax method integrates classical techniques with modern symbolic computation and exhibits strong compatibility with algebraic solvers like Maple. Graphical simulations of the resulting wave profiles validate the accuracy and diversity of the solutions and highlight the potential of the method for applications in nonlinear wave theory, soliton dynamics, and computational mathematics. Compared to existing methods, the softmax approach provides an elegant, scalable, and exact mechanism for solving high\u2010dimensional nonlinear systems. This framework may be extended to broader classes of integrable and nonintegrable PDEs in future work.<\/jats:p>","DOI":"10.1155\/jama\/8810599","type":"journal-article","created":{"date-parts":[[2025,8,16]],"date-time":"2025-08-16T07:03:53Z","timestamp":1755327833000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["New Solutions of Breaking Soliton Equation Using Softmax Method"],"prefix":"10.1155","volume":"2025","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4035-1759","authenticated-orcid":false,"given":"Nguyen Minh","family":"Tuan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3742-6457","authenticated-orcid":false,"given":"Phayung","family":"Meesad","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-4466-7160","authenticated-orcid":false,"given":"Nguyen Hong","family":"Son","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,8,16]]},"reference":[{"key":"e_1_2_11_1_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2022.105364"},{"key":"e_1_2_11_2_2","doi-asserted-by":"publisher","DOI":"10.3390\/math12132034"},{"key":"e_1_2_11_3_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2021.104793"},{"key":"e_1_2_11_4_2","doi-asserted-by":"publisher","DOI":"10.1088\/1572-9494\/ac11ee"},{"key":"e_1_2_11_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2022.108029"},{"key":"e_1_2_11_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2018.04.011"},{"key":"e_1_2_11_7_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2017.03.001"},{"key":"e_1_2_11_8_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2018.04.013"},{"key":"e_1_2_11_9_2","doi-asserted-by":"publisher","DOI":"10.3934\/math.2024570"},{"key":"e_1_2_11_10_2","doi-asserted-by":"publisher","DOI":"10.1007\/s42286-019-00022-9"},{"key":"e_1_2_11_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rinp.2022.105428"},{"key":"e_1_2_11_12_2","first-page":"69","article-title":"Generalized (2+1)-Dimensional Breaking Soliton Equation","volume":"1","author":"Wazwaz A. M.","year":"2011","journal-title":"TWMS Journal of Applied and Engineering Mathematics"},{"key":"e_1_2_11_13_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2015.05.015"},{"key":"e_1_2_11_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2020.105447"},{"key":"e_1_2_11_15_2","doi-asserted-by":"publisher","DOI":"10.1007\/s13324-018-0258-0"},{"key":"e_1_2_11_16_2","doi-asserted-by":"publisher","DOI":"10.3934\/math.2023908"},{"key":"e_1_2_11_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aej.2020.05.027"},{"key":"e_1_2_11_18_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2015.05.007"},{"key":"e_1_2_11_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2017.08.033"},{"key":"e_1_2_11_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2021.100198"},{"key":"e_1_2_11_21_2","doi-asserted-by":"publisher","DOI":"10.1038\/srep03154"},{"key":"e_1_2_11_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2009.06.066"},{"key":"e_1_2_11_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.joes.2020.11.010"},{"key":"e_1_2_11_24_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2018.12.011"},{"key":"e_1_2_11_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.joes.2021.08.002"},{"key":"e_1_2_11_26_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2009.03.033"},{"key":"e_1_2_11_27_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2009.08.013"},{"key":"e_1_2_11_28_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.08.065"},{"key":"e_1_2_11_29_2","doi-asserted-by":"publisher","DOI":"10.1088\/0256-307X\/28\/9\/090203"},{"key":"e_1_2_11_30_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2017.12.016"},{"key":"e_1_2_11_31_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-021-06864-8"},{"key":"e_1_2_11_32_2","article-title":"Application of the HPM for Nonlinear (3+1)-Dimensional Breaking Soliton Equation","volume":"5","author":"Demiray S. T.","year":"2014","journal-title":"Mathematics in Engineering, Science & Aerospace (MESA)"},{"key":"e_1_2_11_33_2","doi-asserted-by":"publisher","DOI":"10.1080\/16583655.2024.2331984"},{"key":"e_1_2_11_34_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-023-08287-z"},{"key":"e_1_2_11_35_2","doi-asserted-by":"publisher","DOI":"10.1088\/0031-8949\/81\/03\/035005"},{"key":"e_1_2_11_36_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.wavemoti.2023.103243"},{"key":"e_1_2_11_37_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physleta.2019.126178"},{"key":"e_1_2_11_38_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2007.07.043"},{"key":"e_1_2_11_39_2","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(95)00176-Y"},{"key":"e_1_2_11_40_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11424-024-3467-7"},{"key":"e_1_2_11_41_2","first-page":"31","article-title":"Some Exact Solutions of the (2\u2009+\u20091)-Dimensional Breaking Soliton Equation Using the Three-Wave Method","volume":"87","author":"Darvishi M. T.","year":"2012","journal-title":"World Academy of Science, Engineering and Technology"},{"key":"e_1_2_11_42_2","doi-asserted-by":"publisher","DOI":"10.1088\/0256-307X\/24\/2\/001"},{"key":"e_1_2_11_43_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-45528-0"},{"key":"e_1_2_11_44_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-36721-3"},{"key":"e_1_2_11_45_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-79757-7_10"},{"key":"e_1_2_11_46_2","doi-asserted-by":"publisher","DOI":"10.37256\/ccds.3220221488"},{"key":"e_1_2_11_47_2","doi-asserted-by":"publisher","DOI":"10.34257\/GJCSTBVOL22IS1PG47"},{"key":"e_1_2_11_48_2","doi-asserted-by":"publisher","DOI":"10.4208\/eajam.091119.140220"},{"key":"e_1_2_11_49_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2010.05.037"},{"key":"e_1_2_11_50_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2025.116791"},{"key":"e_1_2_11_51_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10773-025-06002-9"},{"key":"e_1_2_11_52_2","doi-asserted-by":"publisher","DOI":"10.69793\/ijmcs\/03.2025\/tuan"},{"key":"e_1_2_11_53_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-031-58561-6_6"},{"key":"e_1_2_11_54_2","doi-asserted-by":"publisher","DOI":"10.22144\/ctujoisd.2023.032"},{"key":"e_1_2_11_55_2","doi-asserted-by":"publisher","DOI":"10.1007\/s42979-023-02339-2"},{"key":"e_1_2_11_56_2","doi-asserted-by":"publisher","DOI":"10.5220\/0011096600003179"},{"key":"e_1_2_11_57_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2008.07.021"},{"key":"e_1_2_11_58_2","doi-asserted-by":"publisher","DOI":"10.3390\/sym15020360"},{"key":"e_1_2_11_59_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2024.100682"},{"key":"e_1_2_11_60_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-023-08343-8"},{"key":"e_1_2_11_61_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11082-023-04792-y"},{"key":"e_1_2_11_62_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11082-023-04549-7"},{"key":"e_1_2_11_63_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.padiff.2024.100969"},{"key":"e_1_2_11_64_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11082-024-06379-7"},{"key":"e_1_2_11_65_2","doi-asserted-by":"publisher","DOI":"10.1515\/nleng-2024-0043"},{"key":"e_1_2_11_66_2","doi-asserted-by":"publisher","DOI":"10.1007\/s12210-024-01284-3"},{"key":"e_1_2_11_67_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11082-023-05425-0"},{"key":"e_1_2_11_68_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.aej.2024.01.058"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/8810599","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/full-xml\/10.1155\/jama\/8810599","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1155\/jama\/8810599","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T02:41:17Z","timestamp":1776393677000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1155\/jama\/8810599"}},"subtitle":[],"editor":[{"given":"Kannan","family":"Krithivasan","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2025,1]]},"references-count":68,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["10.1155\/jama\/8810599"],"URL":"https:\/\/doi.org\/10.1155\/jama\/8810599","archive":["Portico"],"relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"value":"1110-757X","type":"print"},{"value":"1687-0042","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1]]},"assertion":[{"value":"2024-12-10","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-07-16","order":2,"name":"accepted","label":"Accepted","group":{"name":"publication_history","label":"Publication History"}},{"value":"2025-08-16","order":3,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}],"article-number":"8810599"}}