On the Korteweg-de Vries equation: an associated equation
The purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equation ut−6uux+uxxx=0and another nonlinear partial differential equation of the form zt+zxxx−3zxzxxz=H(t)z.The second equation will be called the Associated Equation (AE) and the connection between the two...
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| Format: | Article |
| Language: | English |
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Wiley
1984-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171284000272 |
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| _version_ | 1832552128985432064 |
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| author | Eugene P. Schlereth Ervin Y. Rodin |
| author_facet | Eugene P. Schlereth Ervin Y. Rodin |
| author_sort | Eugene P. Schlereth |
| collection | DOAJ |
| description | The purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equation ut−6uux+uxxx=0and another nonlinear partial differential equation of the form zt+zxxx−3zxzxxz=H(t)z.The second equation will be called the Associated Equation (AE) and the connection between the two will be explained. By considering AE, explicit solutions to KdV will be obtained. These solutions include the solitary wave and the cnoidal wave solutions. In addition, similarity solutions in terms of Airy functions and Painlevé transcendents are found. The approach here is different from the Inverse Scattering Transform and the results are not in the form of solutions to specific initial value problems, but rather in terms of solutions containing arbitrary constants. |
| format | Article |
| id | doaj-art-fbc31465e12d4152bfe67db4a1ee83a0 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1984-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-fbc31465e12d4152bfe67db4a1ee83a02025-02-03T05:59:29ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251984-01-017226327710.1155/S0161171284000272On the Korteweg-de Vries equation: an associated equationEugene P. Schlereth0Ervin Y. Rodin1Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga 37402, Tennessee, USADepartment of Systems Science and Mathematics, Washington University, St. Louis 63130, Missouri, USAThe purpose of this paper is to describe a relationship between the Korteweg-de Vries (KdV) equation ut−6uux+uxxx=0and another nonlinear partial differential equation of the form zt+zxxx−3zxzxxz=H(t)z.The second equation will be called the Associated Equation (AE) and the connection between the two will be explained. By considering AE, explicit solutions to KdV will be obtained. These solutions include the solitary wave and the cnoidal wave solutions. In addition, similarity solutions in terms of Airy functions and Painlevé transcendents are found. The approach here is different from the Inverse Scattering Transform and the results are not in the form of solutions to specific initial value problems, but rather in terms of solutions containing arbitrary constants.http://dx.doi.org/10.1155/S0161171284000272Korteweg-de Vries equationPainlevé transcendentsolitary waves. |
| spellingShingle | Eugene P. Schlereth Ervin Y. Rodin On the Korteweg-de Vries equation: an associated equation International Journal of Mathematics and Mathematical Sciences Korteweg-de Vries equation Painlevé transcendent solitary waves. |
| title | On the Korteweg-de Vries equation: an associated equation |
| title_full | On the Korteweg-de Vries equation: an associated equation |
| title_fullStr | On the Korteweg-de Vries equation: an associated equation |
| title_full_unstemmed | On the Korteweg-de Vries equation: an associated equation |
| title_short | On the Korteweg-de Vries equation: an associated equation |
| title_sort | on the korteweg de vries equation an associated equation |
| topic | Korteweg-de Vries equation Painlevé transcendent solitary waves. |
| url | http://dx.doi.org/10.1155/S0161171284000272 |
| work_keys_str_mv | AT eugenepschlereth onthekortewegdevriesequationanassociatedequation AT ervinyrodin onthekortewegdevriesequationanassociatedequation |