Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods
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 Hiro...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-12-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/23/1/10 |
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| author | Nikolay K. Vitanov Zlatinka I. Dimitrova Kaloyan N. Vitanov |
| author_facet | Nikolay K. Vitanov Zlatinka I. Dimitrova Kaloyan N. Vitanov |
| author_sort | Nikolay K. Vitanov |
| collection | DOAJ |
| description | 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 Schrödinger 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 “small” parameter <inline-formula><math display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>; (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 <inline-formula><math display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula> to solution for arbitrary finite values of <inline-formula><math display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>. 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. |
| first_indexed | 2024-03-10T13:49:10Z |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T13:49:10Z |
| publishDate | 2020-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-2fba7f2acfe847a6a5e5f364a80cd2ce2023-11-21T02:19:34ZengMDPI AGEntropy1099-43002020-12-012311010.3390/e23010010Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other MethodsNikolay K. Vitanov0Zlatinka I. Dimitrova1Kaloyan N. Vitanov2Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, BulgariaInstitute of Solid State Physics, Bulgarian Academy of Sciences, Blvd. Tzarigradsko Chaussee 72, 1784 Sofia, BulgariaInstitute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Block 4, 1113 Sofia, BulgariaThe 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 Schrödinger 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 “small” parameter <inline-formula><math display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>; (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 <inline-formula><math display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula> to solution for arbitrary finite values of <inline-formula><math display="inline"><semantics><mi>ϵ</mi></semantics></math></inline-formula>. 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.https://www.mdpi.com/1099-4300/23/1/10nonlinear partial differential equationsexact solutionsSimple Equations Method (SEsM)Hirota methodinverse scattering transform methodhomogeneous balance method |
| spellingShingle | Nikolay K. Vitanov Zlatinka I. Dimitrova Kaloyan N. Vitanov Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods Entropy nonlinear partial differential equations exact solutions Simple Equations Method (SEsM) Hirota method inverse scattering transform method homogeneous balance method |
| title | Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods |
| title_full | Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods |
| title_fullStr | Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods |
| title_full_unstemmed | Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods |
| title_short | Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods |
| title_sort | simple equations method sesm algorithm connection with hirota method inverse scattering transform method and several other methods |
| topic | nonlinear partial differential equations exact solutions Simple Equations Method (SEsM) Hirota method inverse scattering transform method homogeneous balance method |
| url | https://www.mdpi.com/1099-4300/23/1/10 |
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