{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T04:50:22Z","timestamp":1763959822822,"version":"build-2065373602"},"reference-count":82,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T00:00:00Z","timestamp":1756339200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T00:00:00Z","timestamp":1756339200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Evol. Intel."],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s12065-025-01068-0","type":"journal-article","created":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T02:22:14Z","timestamp":1756347734000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On the analytical solution of the nonlinear strain wave equation through the bilinear neural network method"],"prefix":"10.1007","volume":"18","author":[{"given":"Humaira","family":"Muslim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nauman","family":"Ahmed","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Siegfried","family":"Mac\u00edas","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad Ozair","family":"Ahmed","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad Z.","family":"Baber","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Baboucarr","family":"Ceesay","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jorge\u00a0E.","family":"Mac\u00edas-D\u00edaz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,28]]},"reference":[{"issue":"10","key":"1068_CR1","doi-asserted-by":"crossref","first-page":"1543","DOI":"10.3390\/math12101543","volume":"12","author":"M De Falco","year":"2024","unstructured":"De Falco M, Rarit\u00e1 L, Vaccaro A (2024) A fluid dynamic approach to model and optimize energy flows in networked systems. Mathematics 12(10):1543","journal-title":"Mathematics"},{"issue":"10","key":"1068_CR2","doi-asserted-by":"crossref","first-page":"194","DOI":"10.3390\/computation12100194","volume":"12","author":"MP D\u2019Arienzo","year":"2024","unstructured":"D\u2019Arienzo MP, Rarit\u00e1 L (2024) Dynamics of blood flows in the cardiocirculatory system. Computation 12(10):194","journal-title":"Computation"},{"key":"1068_CR3","doi-asserted-by":"crossref","unstructured":"Briani M, Manzo R, Piccoli B, Rarit\u00e1 L (2024) Estimation of NO $$_ {x}$$ and O $$_ {3}$$ reduction by dissipating traffic waves. Netw Heterogeneous Media, 19(2)","DOI":"10.3934\/nhm.2024037"},{"issue":"1","key":"1068_CR4","doi-asserted-by":"crossref","first-page":"20240126","DOI":"10.1515\/math-2024-0126","volume":"23","author":"M Briani","year":"2025","unstructured":"Briani M, Denaro CA, Piccoli B, Rarit\u00e1 L (2025) Dynamics of particulate emissions in the presence of autonomous vehicles. Open Math 23(1):20240126","journal-title":"Open Math"},{"key":"1068_CR5","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1007\/s11071-018-4387-7","volume":"94","author":"X Liu","year":"2018","unstructured":"Liu X, Triki H, Zhou Q, Liu W, Biswas A (2018) Analytic study on interactions between periodic solitons with controllable parameters. Nonlinear Dyn 94:703\u2013709","journal-title":"Nonlinear Dyn"},{"key":"1068_CR6","doi-asserted-by":"crossref","first-page":"2533","DOI":"10.1007\/s11071-018-4340-9","volume":"93","author":"Y Zhang","year":"2018","unstructured":"Zhang Y, Liu Y, Tang X (2018) M-lump and interactive solutions to a (3++ 1)-dimensional nonlinear system. Nonlinear Dyn 93:2533\u20132541","journal-title":"Nonlinear Dyn"},{"key":"1068_CR7","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1007\/s11071-018-4428-2","volume":"94","author":"Y Zhang","year":"2018","unstructured":"Zhang Y, Yang C, Yu W, Mirzazadeh M, Zhou Q, Liu W (2018) Interactions of vector anti-dark solitons for the coupled nonlinear Schr\u00f6dinger equation in inhomogeneous fibers. Nonlinear Dyn 94:1351\u20131360","journal-title":"Nonlinear Dyn"},{"key":"1068_CR8","doi-asserted-by":"crossref","first-page":"1931","DOI":"10.1007\/s11071-017-3991-2","volume":"91","author":"A Ankiewicz","year":"2018","unstructured":"Ankiewicz A, Akhmediev N (2018) Rogue wave-type solutions of the mKdV equation and their relation to known NLSE rogue wave solutions. Nonlinear Dyn 91:1931\u20131938","journal-title":"Nonlinear Dyn"},{"key":"1068_CR9","volume":"42","author":"S Yu","year":"2022","unstructured":"Yu S, Huang L (2022) Exact solutions of the generalized (2+ 1)-dimensional shallow water wave equation. Results Phys 42:106020","journal-title":"Results Phys"},{"key":"1068_CR10","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s11071-015-2305-9","volume":"83","author":"B Carboni","year":"2016","unstructured":"Carboni B, Lacarbonara W (2016) Nonlinear dynamic characterization of a new hysteretic device: experiments and computations. Nonlinear Dyn 83:23\u201339","journal-title":"Nonlinear Dyn"},{"issue":"9","key":"1068_CR11","doi-asserted-by":"crossref","first-page":"2334","DOI":"10.1016\/j.camwa.2016.08.027","volume":"72","author":"Y Tang","year":"2016","unstructured":"Tang Y, Tao S, Guan Q (2016) Lump solitons and the interaction phenomena of them for two classes of nonlinear evolution equations. Comput Math Appl 72(9):2334\u20132342","journal-title":"Comput Math Appl"},{"issue":"06","key":"1068_CR12","doi-asserted-by":"crossref","first-page":"1850074","DOI":"10.1142\/S0217984918500744","volume":"32","author":"R Zhang","year":"2018","unstructured":"Zhang R, Bilige S, Bai Y, L\u00fc J, Gao X (2018) Interaction phenomenon to dimensionally reduced p-gBKP equation. Mod Phys Lett B 32(06):1850074","journal-title":"Mod Phys Lett B"},{"key":"1068_CR13","doi-asserted-by":"crossref","first-page":"2487","DOI":"10.1007\/s11071-012-0636-3","volume":"70","author":"A Arena","year":"2012","unstructured":"Arena A, Lacarbonara W (2012) Nonlinear parametric modeling of suspension bridges under aeroelastic forces: torsional divergence and flutter. Nonlinear Dyn 70:2487\u20132510","journal-title":"Nonlinear Dyn"},{"key":"1068_CR14","doi-asserted-by":"crossref","first-page":"1669","DOI":"10.1007\/s11071-017-3972-5","volume":"91","author":"J L\u00fc","year":"2018","unstructured":"L\u00fc J, Bilige S, Chaolu T (2018) The study of lump solution and interaction phenomenon to (2+ 1)(2+ 1)-dimensional generalized fifth-order KdV equation. Nonlinear Dyn 91:1669\u20131676","journal-title":"Nonlinear Dyn"},{"key":"1068_CR15","doi-asserted-by":"crossref","first-page":"1669","DOI":"10.1007\/s11071-017-3972-5","volume":"91","author":"J L\u00fc","year":"2018","unstructured":"L\u00fc J, Bilige S, Chaolu T (2018) The study of lump solution and interaction phenomenon to (2+ 1)(2+ 1)-dimensional generalized fifth-order KdV equation. Nonlinear Dyn 91:1669\u20131676","journal-title":"Nonlinear Dyn"},{"issue":"2","key":"1068_CR16","doi-asserted-by":"crossref","first-page":"5851","DOI":"10.29020\/nybg.ejpam.v18i2.5851","volume":"18","author":"B Ceesay","year":"2025","unstructured":"Ceesay B, Baber MZ, Ahmed N, Jawaz M, Mac\u00edas-D\u00edaz JE, Gallegos A (2025) Harvesting mixed, homoclinic breather, M-shaped, and other wave profiles of the Heisenberg Ferromagnet-type Akbota equation. Eur J Pure Appl Math 18(2):5851\u20135851","journal-title":"Eur J Pure Appl Math"},{"key":"1068_CR17","doi-asserted-by":"crossref","unstructured":"Ozsahin DU, Ceesay B, Baber MZ, Ahmed N, Raza A, Rafiq M, Ismail EA (2024) Multiwaves, breathers, lump and other solutions for the Heimburg model in biomembranes and nerves. Sci Rep 14(1):10180","DOI":"10.1038\/s41598-024-60689-0"},{"issue":"1","key":"1068_CR18","doi-asserted-by":"crossref","first-page":"20240057","DOI":"10.1515\/phys-2024-0057","volume":"22","author":"B Ceesay","year":"2024","unstructured":"Ceesay B, Ahmed N, Mac\u00edas-D\u00edaz JE (2024) Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota\u2019s bilinear method. Open Phys 22(1):20240057","journal-title":"Open Phys"},{"issue":"04","key":"1068_CR19","doi-asserted-by":"crossref","first-page":"2350247","DOI":"10.1142\/S0217984923502470","volume":"38","author":"S Behera","year":"2024","unstructured":"Behera S (2024) Analysis of traveling wave solutions of two space-time nonlinear fractional differential equations by the first-integral method. Mod Phys Lett B 38(04):2350247","journal-title":"Mod Phys Lett B"},{"issue":"3","key":"1068_CR20","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1007\/s12043-023-02602-4","volume":"97","author":"S Behera","year":"2023","unstructured":"Behera S, Aljahdaly NH (2023) Nonlinear evolution equations and their traveling wave solutions in fluid media by modified analytical method. Pramana 97(3):130","journal-title":"Pramana"},{"key":"1068_CR21","volume":"8","author":"S Behera","year":"2023","unstructured":"Behera S, Mohanty S, Virdi JPS (2023) Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations. Partial Differ Equ Appl Math 8:100535","journal-title":"Partial Differ Equ Appl Math"},{"issue":"5","key":"1068_CR22","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/s10773-024-05647-2","volume":"63","author":"S Behera","year":"2024","unstructured":"Behera S, Aljahdaly NH (2024) Soliton solutions of nonlinear geophysical KdV equation via two analytical methods. Int J Theor Phys 63(5):107","journal-title":"Int J Theor Phys"},{"issue":"01","key":"1068_CR23","doi-asserted-by":"crossref","first-page":"2450403","DOI":"10.1142\/S0217984924504037","volume":"39","author":"S Behera","year":"2025","unstructured":"Behera S (2025) Optical solitons for the Hirota\u2013Ramani equation via improved $$\\frac{G^{\\prime }}{G}$$-expansion method. Mod Phys Lett B 39(01):2450403","journal-title":"Mod Phys Lett B"},{"issue":"7","key":"1068_CR24","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1007\/s11082-024-06403-w","volume":"56","author":"S Behera","year":"2024","unstructured":"Behera S (2024) Multiple soliton solutions of some conformable fractional nonlinear models using Sine\u2013Cosine method. Opt Quant Electron 56(7):1235","journal-title":"Opt Quant Electron"},{"issue":"3","key":"1068_CR25","doi-asserted-by":"crossref","first-page":"458","DOI":"10.1007\/s11766-024-4643-1","volume":"39","author":"S Behera","year":"2024","unstructured":"Behera S, Virdi JSP (2024) Analytical solutions fractional order partial differential equations arising in fluid dynamics. Appl Math-A J Chin Univ 39(3):458\u2013468","journal-title":"Appl Math-A J Chin Univ"},{"key":"1068_CR26","doi-asserted-by":"crossref","unstructured":"Shakeel M, Liu X (2024) Solitary wave solutions, bifurcation theory, and sensitivity analysis in the modified unstable nonlinear Schr\u00f6dinger equation. Mod Phys Lett B, 2550082","DOI":"10.1142\/S0217984925500824"},{"issue":"8","key":"1068_CR27","doi-asserted-by":"crossref","first-page":"8951","DOI":"10.1007\/s11071-024-10757-x","volume":"113","author":"M Shakeel","year":"2025","unstructured":"Shakeel M, Liu X, Abbas N (2025) Investigation of nonlinear dynamics in the stochastic nonlinear Schr\u00f6 dinger equation with spatial noise intensity. Nonlinear Dyn 113(8):8951\u20138971","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"1068_CR28","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1007\/s44198-024-00231-y","volume":"31","author":"M Shakeel","year":"2024","unstructured":"Shakeel M, Liu X, Mostafa AM, AlQahtani NF, Alameri A (2024) Dynamic solitary wave solutions arising in nonlinear chains of atoms model. J Nonlinear Math Phys 31(1):70","journal-title":"J Nonlinear Math Phys"},{"issue":"1","key":"1068_CR29","doi-asserted-by":"crossref","first-page":"12844","DOI":"10.1038\/s41598-024-62315-5","volume":"14","author":"M Shakeel","year":"2024","unstructured":"Shakeel M, Abbas N, Rehman MJU, Alshammari FS, Al-Yaari A (2024) Lie symmetry analysis and solitary wave solution of biofilm model Allen\u2013Cahn. Sci Rep 14(1):12844","journal-title":"Sci Rep"},{"issue":"02","key":"1068_CR30","doi-asserted-by":"crossref","first-page":"2450390","DOI":"10.1142\/S0217984924503901","volume":"39","author":"N Abbas","year":"2025","unstructured":"Abbas N, Shakeel M, Fouly A, Ahmadian H (2025) Numerical simulations and analytical approach for three-component coupled NLS-type equations in fiber optics. Mod Phys Lett B 39(02):2450390","journal-title":"Mod Phys Lett B"},{"issue":"2","key":"1068_CR31","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1007\/s11071-015-2539-6","volume":"84","author":"WX Ma","year":"2016","unstructured":"Ma WX, Qin Z, L\u00fc X (2016) Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn 84(2):923\u2013931","journal-title":"Nonlinear Dyn"},{"issue":"4","key":"1068_CR32","doi-asserted-by":"crossref","first-page":"2633","DOI":"10.1016\/j.jde.2017.10.033","volume":"264","author":"WX Ma","year":"2018","unstructured":"Ma WX, Zhou Y (2018) Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differ Equ 264(4):2633\u20132659","journal-title":"J Differ Equ"},{"issue":"3","key":"1068_CR33","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1016\/j.camwa.2017.05.010","volume":"74","author":"JB Zhang","year":"2017","unstructured":"Zhang JB, Ma WX (2017) Mixed lump-kink solutions to the BKP equation. Comput Math Appl 74(3):591\u2013596","journal-title":"Comput Math Appl"},{"key":"1068_CR34","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s13324-017-0181-9","volume":"8","author":"JY Yang","year":"2018","unstructured":"Yang JY, Ma WX, Qin Z (2018) Lump and lump-soliton solutions to the (2+ 1)(2+ 1)-dimensional Ito equation. Anal Math Phys 8:427\u2013436","journal-title":"Anal Math Phys"},{"issue":"1","key":"1068_CR35","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.camwa.2017.09.013","volume":"75","author":"WX Ma","year":"2018","unstructured":"Ma WX, Yong X, Zhang HQ (2018) Diversity of interaction solutions to the (2+ 1)-dimensional Ito equation. Comput Math Appl 75(1):289\u2013295","journal-title":"Comput Math Appl"},{"issue":"4","key":"1068_CR36","first-page":"140","volume":"2","author":"WX Ma","year":"2011","unstructured":"Ma WX (2011) Generalized bilinear differential equations. Stud Nonlinear Sci 2(4):140\u2013144","journal-title":"Stud Nonlinear Sci"},{"issue":"36","key":"1068_CR37","doi-asserted-by":"crossref","first-page":"1975","DOI":"10.1016\/j.physleta.2015.06.061","volume":"379","author":"WX Ma","year":"2015","unstructured":"Ma WX (2015) Lump solutions to the Kadomtsev\u2013Petviashvili equation. Phys Lett A 379(36):1975\u20131978","journal-title":"Phys Lett A"},{"key":"1068_CR38","doi-asserted-by":"crossref","unstructured":"Ma WX (2013) Bilinear equations, Bell polynomials and linear superposition principle. J Phys: Conf Ser 411(1):012021. IOP Publishing","DOI":"10.1088\/1742-6596\/411\/1\/012021"},{"issue":"1","key":"1068_CR39","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.camwa.2017.09.013","volume":"75","author":"WX Ma","year":"2018","unstructured":"Ma WX, Yong X, Zhang HQ (2018) Diversity of interaction solutions to the (2+ 1)-dimensional Ito equation. Comput Math Appl 75(1):289\u2013295","journal-title":"Comput Math Appl"},{"key":"1068_CR40","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/j.aml.2015.06.003","volume":"50","author":"X L\u00fc","year":"2015","unstructured":"L\u00fc X, Ma WX, Khalique CM (2015) A direct bilinear B\u00e4cklund transformation of a (2+ 1)-dimensional Korteweg\u2013de Vries-like model. Appl Math Lett 50:37\u201342","journal-title":"Appl Math Lett"},{"issue":"3","key":"1068_CR41","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1080\/17455030.2017.1367866","volume":"28","author":"Y Sun","year":"2018","unstructured":"Sun Y, Tian B, Xie XY, Chai J, Yin HM (2018) Rogue waves and lump solitons for a-dimensional B-type Kadomtsev\u2013Petviashvili equation in fluid dynamics. Waves Random Complex Media 28(3):544\u2013552","journal-title":"Waves Random Complex Media"},{"key":"1068_CR42","doi-asserted-by":"crossref","first-page":"1511","DOI":"10.1007\/s13324-018-0258-0","volume":"9","author":"MJ Dong","year":"2019","unstructured":"Dong MJ, Tian SF, Wang XB, Zhang TT (2019) Lump-type solutions and interaction solutions in the (3+ 1)-dimensional potential Yu\u2013Toda\u2013Sasa\u2013Fukuyama equation. Anal Math Phys 9:1511\u20131523","journal-title":"Anal Math Phys"},{"key":"1068_CR43","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.spmi.2016.12.019","volume":"102","author":"SL Jia","year":"2017","unstructured":"Jia SL, Gao YT, Hu L, Huang QM, Hu WQ (2017) Soliton-like, periodic wave and rational solutions for a (3+ 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation in the incompressible fluid. Superlattices Microstruct 102:273\u2013283","journal-title":"Superlattices Microstruct"},{"key":"1068_CR44","doi-asserted-by":"crossref","first-page":"523","DOI":"10.1007\/s11071-016-2905-z","volume":"86","author":"X L\u00fc","year":"2016","unstructured":"L\u00fc X, Chen ST, Ma WX (2016) Constructing lump solutions to a generalized Kadomtsev\u2013Petviashvili\u2013Boussinesq equation. Nonlinear Dyn 86:523\u2013534","journal-title":"Nonlinear Dyn"},{"issue":"5","key":"1068_CR45","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1016\/j.camwa.2016.06.008","volume":"72","author":"LN Gao","year":"2016","unstructured":"Gao LN, Zhao XY, Zi YY, Yu J, L\u00fc X (2016) Resonant behavior of multiple wave solutions to a Hirota bilinear equation. Comput Math Appl 72(5):1225\u20131229","journal-title":"Comput Math Appl"},{"key":"1068_CR46","doi-asserted-by":"crossref","unstructured":"Gao LN, Zi YY, Yin YH, Ma W, X, L\u00fc X (2017) B\u00e4cklund transformation, multiple wave solutions and lump solutions to a (3+ 1)-dimensional nonlinear evolution equation. Nonlinear Dyn 89(3):2233\u20132240","DOI":"10.1007\/s11071-017-3581-3"},{"key":"1068_CR47","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.cnsns.2015.08.008","volume":"32","author":"X L\u00fc","year":"2016","unstructured":"L\u00fc X, Lin F (2016) Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order. Commun Nonlinear Sci Numer Simul 32:241\u2013261","journal-title":"Commun Nonlinear Sci Numer Simul"},{"key":"1068_CR48","doi-asserted-by":"crossref","unstructured":"Lin FH, Chen ST, Qu QX, Wang JP, Zhou X, W, L\u00fc X (2018) Resonant multiple wave solutions to a new (3+ 1)-dimensional generalized Kadomtsev\u2013Petviashvili equation: linear superposition principle. Appl Math Lett 78:112\u2013117","DOI":"10.1016\/j.aml.2017.10.013"},{"key":"1068_CR49","doi-asserted-by":"crossref","first-page":"1249","DOI":"10.1007\/s11071-017-3942-y","volume":"91","author":"X L\u00fc","year":"2018","unstructured":"L\u00fc X, Wang JP, Lin FH, Zhou XW (2018) Lump dynamics of a generalized two-dimensional Boussinesq equation in shallow water. Nonlinear Dyn 91:1249\u20131259","journal-title":"Nonlinear Dyn"},{"issue":"6","key":"1068_CR50","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1016\/j.camwa.2018.06.020","volume":"76","author":"YH Yin","year":"2018","unstructured":"Yin YH, Ma WX, Liu JG, L\u00fc X (2018) Diversity of exact solutions to a (3+ 1)-dimensional nonlinear evolution equation and its reduction. Comput Math Appl 76(6):1275\u20131283","journal-title":"Comput Math Appl"},{"key":"1068_CR51","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/j.aml.2018.07.001","volume":"86","author":"X Yong","year":"2018","unstructured":"Yong X, Li X, Huang Y (2018) General lump-type solutions of the (3+ 1)-dimensional Jimbo\u2013Miwa equation. Appl Math Lett 86:222\u2013228","journal-title":"Appl Math Lett"},{"key":"1068_CR52","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1007\/s11071-016-3267-2","volume":"88","author":"JG Liu","year":"2017","unstructured":"Liu JG, Du JQ, Zeng ZF, Nie B (2017) New three-wave solutions for the (3+ 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation. Nonlinear Dyn 88:655\u2013661","journal-title":"Nonlinear Dyn"},{"key":"1068_CR53","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s11071-017-3884-4","volume":"91","author":"YZ Li","year":"2018","unstructured":"Li YZ, Liu JG (2018) New periodic solitary wave solutions for the new (2+ 1)-dimensional Korteweg\u2013de Vries equation. Nonlinear Dyn 91:497\u2013504","journal-title":"Nonlinear Dyn"},{"key":"1068_CR54","doi-asserted-by":"crossref","first-page":"24","DOI":"10.1016\/j.cnsns.2017.03.021","volume":"52","author":"X Zhang","year":"2017","unstructured":"Zhang X, Chen Y (2017) Rogue wave and a pair of resonance stripe solitons to a reduced (3+ 1)-dimensional Jimbo\u2013Miwa equation. Commun Nonlinear Sci Numer Simul 52:24\u201331","journal-title":"Commun Nonlinear Sci Numer Simul"},{"issue":"7","key":"1068_CR55","doi-asserted-by":"crossref","first-page":"7121","DOI":"10.1007\/s11071-024-10715-7","volume":"113","author":"S Zhang","year":"2025","unstructured":"Zhang S, Zhu G, Huang W, Wang H, Yang C, Lin Y (2025) Symbolic computation of analytical solutions for nonlinear partial differential equations based on bilinear neural network method. Nonlinear Dyn 113(7):7121\u20137137","journal-title":"Nonlinear Dyn"},{"issue":"15","key":"1068_CR56","doi-asserted-by":"crossref","first-page":"13361","DOI":"10.1007\/s11071-024-09708-3","volume":"112","author":"MA Isah","year":"2024","unstructured":"Isah MA, Yokus A, Kaya D (2024) Exploring the influence of layer and neuron configurations on Boussinesq equation solutions via a bilinear neural network framework. Nonlinear Dyn 112(15):13361\u201313377","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"1068_CR57","doi-asserted-by":"crossref","first-page":"122","DOI":"10.1007\/s11424-020-9392-5","volume":"34","author":"R Zhang","year":"2021","unstructured":"Zhang R, Bilige S, Chaolu T (2021) Fractal solitons, arbitrary function solutions, exact periodic wave and breathers for a nonlinear partial differential equation by using bilinear neural network method. J Syst Sci Complexity 34(1):122\u2013139","journal-title":"J Syst Sci Complexity"},{"key":"1068_CR58","doi-asserted-by":"crossref","first-page":"5612","DOI":"10.1002\/mma.8131","volume":"9","author":"JM Qiao","year":"2022","unstructured":"Qiao JM, Zhang RF, Yue RX, Rezazadeh H, Seadawy AR (2022) Three types of periodic solutions of new (3+ 1)-dimensional Boiti\u2013Leon\u2013Manna\u2013Pempinelli equation via bilinear neural network method. Math Methods Appl Sci 9:5612\u20135621","journal-title":"Math Methods Appl Sci"},{"key":"1068_CR59","doi-asserted-by":"crossref","first-page":"3041","DOI":"10.1007\/s11071-018-04739-z","volume":"95","author":"RF Zhang","year":"2019","unstructured":"Zhang RF, Bilige S (2019) Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn 95:3041\u20133048","journal-title":"Nonlinear Dyn"},{"key":"1068_CR60","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2021.111692","volume":"154","author":"RF Zhang","year":"2022","unstructured":"Zhang RF, Li MC, Gan JY, Li Q, Lan ZZ (2022) Novel trial functions and rogue waves of generalized breaking soliton equation via bilinear neural network method. Chaos Solitons Fractals 154:111692","journal-title":"Chaos Solitons Fractals"},{"issue":"2","key":"1068_CR61","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1007\/s40819-025-01851-8","volume":"11","author":"NM Tuan","year":"2025","unstructured":"Tuan NM, Meesad P (2025) Bilinear recurrent neural network for a modified Benney\u2013Luke equation. Int J Appl Comput Math 11(2):35","journal-title":"Int J Appl Comput Math"},{"issue":"7","key":"1068_CR62","doi-asserted-by":"crossref","first-page":"7185","DOI":"10.1007\/s11071-024-10495-0","volume":"113","author":"Y Ye","year":"2025","unstructured":"Ye Y, Fan H, Li Y, Liu X, Zhang H (2025) Conformable bilinear neural network method: a novel method for time-fractional nonlinear partial differential equations in the sense of conformable derivative. Nonlinear Dyn 113(7):7185\u20137200","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"1068_CR63","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s11071-022-07207-x","volume":"108","author":"RF Zhang","year":"2022","unstructured":"Zhang RF, Li MC (2022) Bilinear residual network method for solving the exactly explicit solutions of nonlinear evolution equations. Nonlinear Dyn 108(1):521\u2013531","journal-title":"Nonlinear Dyn"},{"key":"1068_CR64","doi-asserted-by":"crossref","unstructured":"Wu X, S, Liu JG (2024) Solving the variable coefficient nonlinear partial differential equations based on the bilinear residual network method. Nonlinear Dyn 112(10):8329\u20138340","DOI":"10.1007\/s11071-024-09472-4"},{"issue":"2\u20133","key":"1068_CR65","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/0022-3093(90)90703-O","volume":"116","author":"GP Johari","year":"1990","unstructured":"Johari GP, Ram S, Astl G, Mayer E (1990) Characterizing amorphous and microcrystalline solids by calorimetry. J Non-Cryst Solids 116(2\u20133):282\u2013285","journal-title":"J Non-Cryst Solids"},{"issue":"2\u20133","key":"1068_CR66","doi-asserted-by":"crossref","first-page":"282","DOI":"10.1016\/0022-3093(90)90703-O","volume":"116","author":"GP Johari","year":"1990","unstructured":"Johari GP, Ram S, Astl G, Mayer E (1990) Characterizing amorphous and microcrystalline solids by calorimetry. J Non-Cryst Solids 116(2\u20133):282\u2013285","journal-title":"J Non-Cryst Solids"},{"key":"1068_CR67","doi-asserted-by":"crossref","unstructured":"Struszczyk H (1987) Microcrystalline chitosan. I. Preparation and properties of microcrystalline chitosan. J Appl Polym Sci 33(1):177\u2013189","DOI":"10.1002\/app.1987.070330115"},{"key":"1068_CR68","doi-asserted-by":"crossref","DOI":"10.1016\/j.jpba.2023.115609","volume":"235","author":"RM Abdelhameed","year":"2023","unstructured":"Abdelhameed RM, Hammad SF, Abdallah IA, Bedair A, Locatelli M, Mansour FR (2023) A hybrid microcrystalline cellulose\/metal-organic framework for dispersive solid phase microextraction of selected pharmaceuticals: A proof-of-concept. J Pharm Biomed Anal 235:115609","journal-title":"J Pharm Biomed Anal"},{"issue":"1","key":"1068_CR69","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1515\/nleng-2016-0020","volume":"6","author":"Z Ayati","year":"2017","unstructured":"Ayati Z, Hosseini K, Mirzazadeh M (2017) Application of Kudryashov and functional variable methods to the strain wave equation in microstructured solids. Nonlinear Eng 6(1):25\u201329","journal-title":"Nonlinear Eng"},{"key":"1068_CR70","doi-asserted-by":"crossref","unstructured":"Wei X, Wen W, Shi W, Liu Y, Sun J, Dai X, Liu Y (2024) Intrinsically stretchable light-emitting polymer semiconductors with high charge mobility through micro-crystalline aggregation-limited morphology. Adv Funct Mater 34(10):2310558","DOI":"10.1002\/adfm.202310558"},{"key":"1068_CR71","doi-asserted-by":"crossref","unstructured":"Leto JA, Choudhury SR (2009) Solitary wave families of a generalized microstructure PDE. Commun Nonlinear Sci Numer Simul 14(5):1999\u20132005","DOI":"10.1016\/j.cnsns.2008.04.016"},{"issue":"11","key":"1068_CR72","doi-asserted-by":"crossref","first-page":"2300189","DOI":"10.1002\/qute.202300189","volume":"6","author":"H Sun","year":"2023","unstructured":"Sun H, Zhang Z, Liu Y, Chen G, Li T, Liao M (2023) Diamond MEMS: from classical to quantum. Adv Quantum Technol 6(11):2300189","journal-title":"Adv Quantum Technol"},{"key":"1068_CR73","doi-asserted-by":"crossref","unstructured":"ur Rehman H, Awan AU, Habib A, Gamaoun F, El Din EMT, Galal AM (2022) Solitary wave solutions for a strain wave equation in a microstructured solid. Results Phys 39:105755","DOI":"10.1016\/j.rinp.2022.105755"},{"issue":"26","key":"1068_CR74","doi-asserted-by":"crossref","first-page":"2150439","DOI":"10.1142\/S021798492150439X","volume":"35","author":"KK Ali","year":"2021","unstructured":"Ali KK, Yilmazer R, Bulut H, Akt\u00fcrk T, Osman MS (2021) Abundant exact solutions to the strain wave equation in micro-structured solids. Mod Phys Lett B 35(26):2150439","journal-title":"Mod Phys Lett B"},{"issue":"3","key":"1068_CR75","doi-asserted-by":"crossref","DOI":"10.1016\/j.asej.2022.101883","volume":"14","author":"M Shakeel","year":"2023","unstructured":"Shakeel M, Shah NA, Chung JD (2023) Application of modified exp-function method for strain wave equation for finding analytical solutions. Ain Shams Eng J 14(3):101883","journal-title":"Ain Shams Eng J"},{"issue":"11","key":"1068_CR76","first-page":"1","volume":"135","author":"S Kumar","year":"2020","unstructured":"Kumar S, Kumar A, Wazwaz AM (2020) New exact solitary wave solutions of the strain wave equation in microstructured solids via the generalized exponential rational function method. Eur Phys J Plus 135(11):1\u201317","journal-title":"Eur Phys J Plus"},{"issue":"27","key":"1068_CR77","doi-asserted-by":"crossref","DOI":"10.1016\/j.physleta.2020.126683","volume":"384","author":"N Raza","year":"2020","unstructured":"Raza N, Seadawy AR, Jhangeer A, Butt AR, Arshed S (2020) Dynamical behavior of micro-structured solids with conformable time fractional strain wave equation. Phys Lett A 384(27):126683","journal-title":"Phys Lett A"},{"issue":"1","key":"1068_CR78","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1515\/nleng-2016-0020","volume":"6","author":"Z Ayati","year":"2017","unstructured":"Ayati Z, Hosseini K, Mirzazadeh M (2017) Application of Kudryashov and functional variable methods to the strain wave equation in microstructured solids. Nonlinear Eng 6(1):25\u201329","journal-title":"Nonlinear Eng"},{"issue":"2","key":"1068_CR79","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1016\/j.asej.2014.11.011","volume":"6","author":"MG Hafez","year":"2015","unstructured":"Hafez MG, Akbar MA (2015) An exponential expansion method and its application to the strain wave equation in microstructured solids. Ain Shams Eng J 6(2):683\u2013690","journal-title":"Ain Shams Eng J"},{"issue":"3","key":"1068_CR80","doi-asserted-by":"crossref","first-page":"650","DOI":"10.3390\/sym15030650","volume":"15","author":"A Aljuaidan","year":"2023","unstructured":"Aljuaidan A, Elbrolosy M, Elmandouh A (2023) Nonlinear wave propagation for a strain wave equation of a flexible rod with finite deformation. Symmetry 15(3):650","journal-title":"Symmetry"},{"key":"1068_CR81","doi-asserted-by":"crossref","unstructured":"Zhestkij N, Povarov S, Mezenov YA, Rzhevskiy SS, Kulakova AN, Milichko VA (2023) UiO-66 metal organic framework as stable platform for non-linear optical applications. Photonics Nanostruct-Fundam Appl 57:101198","DOI":"10.1016\/j.photonics.2023.101198"},{"key":"1068_CR82","doi-asserted-by":"crossref","unstructured":"Yang Z, Wang T, Xu X, Yao J, Xu L, Wang S, Song J (2023) Fiber optic plate coupled Pb-free perovskite X-ray camera featuring low-dose-rate imaging toward dental diagnosis. J Phys Chem Lett 14(2):326\u2013333","DOI":"10.1021\/acs.jpclett.2c03586"}],"container-title":["Evolutionary Intelligence"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12065-025-01068-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12065-025-01068-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12065-025-01068-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T12:17:52Z","timestamp":1761394672000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12065-025-01068-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,28]]},"references-count":82,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["1068"],"URL":"https:\/\/doi.org\/10.1007\/s12065-025-01068-0","relation":{},"ISSN":["1864-5909","1864-5917"],"issn-type":[{"type":"print","value":"1864-5909"},{"type":"electronic","value":"1864-5917"}],"subject":[],"published":{"date-parts":[[2025,8,28]]},"assertion":[{"value":"2 April 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 May 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 June 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 August 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"97"}}