{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T01:44:28Z","timestamp":1775007868962,"version":"3.50.1"},"reference-count":135,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,23]],"date-time":"2020-12-23T00:00:00Z","timestamp":1608681600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Education and Science, Bulgaria","award":["D01205\/23.11.2018"],"award-info":[{"award-number":["D01205\/23.11.2018"]}]},{"name":"Operating Program \u201cScience and Education for Intelligent Growth\u201d of Republic of Bulgaria","award":["BG05 M2OP001-1.001-0008"],"award-info":[{"award-number":["BG05 M2OP001-1.001-0008"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schr\u00f6dinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a \u201csmall\u201d parameter \u03f5; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of \u03f5 to solution for arbitrary finite values of \u03f5. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.<\/jats:p>","DOI":"10.3390\/e23010010","type":"journal-article","created":{"date-parts":[[2020,12,23]],"date-time":"2020-12-23T12:19:51Z","timestamp":1608725991000},"page":"10","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":53,"title":["Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6209-547X","authenticated-orcid":false,"given":"Nikolay K.","family":"Vitanov","sequence":"first","affiliation":[{"name":"Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria"}]},{"given":"Zlatinka I.","family":"Dimitrova","sequence":"additional","affiliation":[{"name":"Institute of Solid State Physics, Bulgarian Academy of Sciences, Blvd. Tzarigradsko Chaussee 72, 1784 Sofia, Bulgaria"}]},{"given":"Kaloyan N.","family":"Vitanov","sequence":"additional","affiliation":[{"name":"Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, Bulgaria"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,23]]},"reference":[{"key":"ref_1","unstructured":"Axelrod, R., and Cohen, M. (2001). Harnessing Complexity, Basic Books."},{"key":"ref_2","unstructured":"Chian, A.C.-L. (2007). Complex Systems Approach to Economic Dynamics, Springer."},{"key":"ref_3","unstructured":"Chen, W.-K. (2003). Theory of Nets. Flows in Networks, Imperial College Press."},{"key":"ref_4","unstructured":"Lucas, M.W. (2010). 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