{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,30]],"date-time":"2026-01-30T07:57:43Z","timestamp":1769759863475,"version":"3.49.0"},"reference-count":39,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,28]],"date-time":"2024-04-28T00:00:00Z","timestamp":1714262400000},"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>This paper uses the attached flow method for solving nonlinear second-order differential equations of the reaction\u2013diffusion type. The key steps of the method consist of the following: (i) reducing the differentiability order by defining the first derivative of the variable as a new variable called the flow and (ii) a forced decomposition of the derivative-free term so that the flow appears explicitly in it. The resulting reduced equation is solved using specific balancing rules. Only step (i) would lead to an Abel-type equation with complicated integral solutions. Completed with (ii) and with the graduation procedure, the attached flow method used in the paper, without requiring such a great effort, allows for the obtaining of accurate analytical solutions. The method is applied here to a subclass of reaction\u2013diffusion equations, the generalized Dodd\u2013Bulough\u2013Mikhailov equation, which includes a translation of the variable and nonlinearities up to order five. The equation is solved for each order of nonlinearity, and the solutions are discussed following the values of the parameters involved in the equation.<\/jats:p>","DOI":"10.3390\/sym16050531","type":"journal-article","created":{"date-parts":[[2024,5,1]],"date-time":"2024-05-01T08:59:39Z","timestamp":1714553979000},"page":"531","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Attached Flows for Reaction\u2013Diffusion Processes Described by a Generalized Dodd\u2013Bullough\u2013Mikhailov Equation"],"prefix":"10.3390","volume":"16","author":[{"given":"Carmen","family":"Ionescu","sequence":"first","affiliation":[{"name":"Department of Physics, University of Craiova, 13 A.I. Cuza, 200585 Craiova, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0875-7879","authenticated-orcid":false,"given":"Iulian","family":"Petrisor","sequence":"additional","affiliation":[{"name":"Department of Physics, University of Craiova, 13 A.I. Cuza, 200585 Craiova, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ionescu, C., and Constantinescu, R. (2022). Solving Nonlinear Second-Order Differential Equations through the Attached Flow Method. Mathematics, 10.","DOI":"10.3390\/math10152811"},{"key":"ref_2","unstructured":"Zwillinger, D. (1997). Handbook of Differential Equations, Academic Press. [3rd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Panayotounakos, D.E., and Zarmpoutis, T.I. (2011). Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel\u2019s Nonlinear ODEs of the First Kind and Relative Degenerate Equations). Int. J. Math. Sci., 387429.","DOI":"10.1155\/2011\/387429"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Agrawal, G.P. (2006). Nonlinear Fiber Optics, Academic Press. [4th ed.].","DOI":"10.1016\/B978-012369516-1\/50011-X"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"072306","DOI":"10.1063\/1.4958809","article-title":"Stochastic field-line wandering in magnetic turbulence with shear: I. Quasi-linear theory","volume":"23","author":"Shalchi","year":"2016","journal-title":"Phys. Plasmas"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"112303","DOI":"10.1063\/1.4996869","article-title":"Stochastic field-line wandering in magnetic turbulence with shear. II. Decorrelation trajectory method","volume":"24","author":"Negrea","year":"2017","journal-title":"Phys. Plasmas"},{"key":"ref_7","unstructured":"Anderson, J.D. (2007). Fundamentals of Aerodynamics, McGraw\u2013Hill. [4th ed.]."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"109566","DOI":"10.1016\/j.jcp.2020.109566","article-title":"On Lagrangian schemes for porous medium type generalized diffusion equations: A discrete energetic variational approach","volume":"417","author":"Liu","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"110253","DOI":"10.1016\/j.jcp.2021.110253","article-title":"A structure-preserving, operator splitting scheme for reaction-diffusion equations with detailed balance","volume":"436","author":"Liu","year":"2021","journal-title":"J. Comput. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Cherniha, R., and Davydovych, V. (2017). Nonlinear Reaction-Diffusion Systems, Springer.","DOI":"10.1007\/978-3-319-65467-6"},{"key":"ref_11","unstructured":"(2003). Handbook of Exact Solutions for Ordinary Differential Equations, Chapman & Hall\/CRC Press."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1016\/0375-9601(88)90027-8","article-title":"Exact and explicit solitary wave solutions for the generalised Fisher equation","volume":"131","author":"Wang","year":"1988","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"3507","DOI":"10.1016\/j.cnsns.2009.01.023","article-title":"Seven common errors in finding exact solutions of nonlinear differential equations","volume":"14","author":"Kudryashov","year":"2009","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"055004","DOI":"10.1088\/2399-6528\/ab1a47","article-title":"Travelling wave solution of Dodd-Bullough-Mikhailov equation: A comparative study between Generalized Kudryashov and improved F-expansion methods","volume":"3","author":"Islam","year":"2019","journal-title":"J. Phys. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Cimpoiasu, R., Constantinescu, R., and Pauna, A.S. (2021). Solutions of the Bullough\u2013Dodd Model of Scalar Field through Jacobi-Type Equations. Symmetry, 13.","DOI":"10.3390\/sym13081529"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1016\/j.nuclphysb.2008.01.004","article-title":"The Bullough-Dodd model coupled to matter fields","volume":"800","author":"Assis","year":"2008","journal-title":"Nucl. Phys. B"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Rui, W. (2013). Exact traveling wave solutions for a nonlinear evolution equation of generalized Tzitz\u00e9ica-Dodd-Bullough-Mikhailov type. J. Appl. Math., 395628.","DOI":"10.1155\/2013\/395628"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"699","DOI":"10.1016\/j.camwa.2010.12.017","article-title":"Traveling wave solutions to the (n+1)-dimensional sinh\u2013cosh\u2013Gordon equation","volume":"61","author":"Fan","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.chaos.2004.09.122","article-title":"The tanh method: Solitons and periodic solutions for the Dodd\u2013Bullough\u2013Mikhailov and the Tzitzeica\u2013Dodd\u2013Bullough equations","volume":"25","author":"Wazwaz","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_20","first-page":"112","article-title":"New solutions of Dodd-Bullough-Mikhailov equation by using an improved tanh-method","volume":"69","author":"Constantinescu","year":"2017","journal-title":"Rom. Rep. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1016\/j.mcm.2011.05.020","article-title":"Application of the (G\u2032\/G)-expansion method for the Zhiber\u2013Shabat equation and other related equations","volume":"54","author":"Borhanifar","year":"2011","journal-title":"Math. Comput. Model."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"569","DOI":"10.11948\/20180314","article-title":"Functional expansions for finding traveling wave solutions","volume":"10","author":"Ionescu","year":"2020","journal-title":"JAAC"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1016\/j.chaos.2006.03.020","article-title":"Exp-function method for nonlinear wave equations","volume":"30","author":"He","year":"2006","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"2248","DOI":"10.1016\/j.cnsns.2011.10.016","article-title":"One method for finding exact solutions of nonlinear differential equations","volume":"17","author":"Kudryashov","year":"2012","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1016\/S0375-9601(02)00626-6","article-title":"A truncated Painlev\u00e9 expansion associated with the Tzitz\u00e9ica equation: Consistency and general solution","volume":"299","author":"Meleshko","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/0375-9601(95)00955-8","article-title":"Darboux transformation and some multi-phase solutions of the Dodd\u2013Bullough\u2013Tzitz\u00e9ica equation","volume":"211","author":"Brezhnev","year":"1996","journal-title":"Phys. Lett. A"},{"key":"ref_27","first-page":"11","article-title":"On Tzitzeica equation and spectral properties of related Lax operators","volume":"19","author":"Babalic","year":"2014","journal-title":"Balk. J. Geom. Its Appl."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2050274","DOI":"10.1142\/S0217979220502744","article-title":"Complete integrability and complex solitons for generalized Volterra system with branched dispersion","volume":"34","author":"Babalic","year":"2020","journal-title":"Int. J. Mod. Phys. B"},{"key":"ref_29","first-page":"114","article-title":"Integrable discretization of coupled Ablowitz-Ladik equations with branched dispersion","volume":"63","author":"Babalic","year":"2018","journal-title":"Rom. J. Phys."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"104583","DOI":"10.1016\/j.rinp.2021.104583","article-title":"Symmetry reductions and invariant-group solutions for a two-dimensional Kundu-Mukherjee-Naskar model","volume":"8","author":"Cimpoiasu","year":"2021","journal-title":"Results Phys."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Shen, G., Sun, Y., and Xiong, Y. (2013). New Travelling-Wave Solutions for Dodd-Bullough Equation. J. Appl. Math., 364718.","DOI":"10.1155\/2013\/364718"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.1016\/j.cjph.2017.07.005","article-title":"Bifurcations of traveling wave solutions for Dodd\u2013Bullough\u2013Mikhailov equation and coupled Higgs equation and their applications","volume":"55","author":"Seadawy","year":"2017","journal-title":"Chin. J. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s11766-007-0004-0","article-title":"Bifurcations of Travelling Wave Solutions For The Generalized Dodd-Bullough-Mikhailov Equation","volume":"22","author":"Wentao","year":"2007","journal-title":"Appl. Math. A J. Chin. Univ. Ser. B"},{"key":"ref_34","unstructured":"Ruffini, P. (1799). Teoria Generale Delle Equazioni, in cui si Dimostra Impossibile la Soluzione Algebraica Delle Equazioni Generali di Grado Superiore al Quarto, Stamperia di S. Tommaso d\u2019Aquino."},{"key":"ref_35","unstructured":"Sylow, L., and Lie, S. (1881). Oeuvres completes de Niels Hendrik Abel I, Grondahl & Son. [2nd ed.]."},{"key":"ref_36","first-page":"508","article-title":"Sur la r\u00e9solution de l\u2019\u00e9quation du cinqui\u00e8me degr\u00e9","volume":"46","author":"Hermite","year":"1858","journal-title":"C. R. Hebd. Seances Acad. Sci."},{"key":"ref_37","first-page":"25","article-title":"Nonlinear dynamical systems in various space\u2013time dimensions","volume":"55","author":"Cimpoiasu","year":"2010","journal-title":"Rom. J. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1002\/andp.20065180302","article-title":"Gauge fixing procedure in the extended BRST theory: The example of the abelian 2\u2013forms","volume":"15","author":"Constantinescu","year":"2006","journal-title":"Ann. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"6629","DOI":"10.1142\/S0217751X06034434","article-title":"Multidifferential complexes and their application to gauge theories","volume":"21","author":"Constantinescu","year":"2006","journal-title":"Int. J. Mod. Phys. A"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/531\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:35:35Z","timestamp":1760106935000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/16\/5\/531"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,28]]},"references-count":39,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["sym16050531"],"URL":"https:\/\/doi.org\/10.3390\/sym16050531","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,28]]}}}