The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three
We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1)...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2018/1642139 |
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| author | Pierre Gaillard |
| author_facet | Pierre Gaillard |
| author_sort | Pierre Gaillard |
| collection | DOAJ |
| description | We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2. |
| format | Article |
| id | doaj-art-a9710dd49dd24944a46e90b45f858b14 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-a9710dd49dd24944a46e90b45f858b142025-02-03T05:47:24ZengWileyAdvances in Mathematical Physics1687-91201687-91392018-01-01201810.1155/2018/16421391642139The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order ThreePierre Gaillard0Université de Bourgogne, Institut de Mathématiques de Bourgogne, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, FranceWe construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N-1 parameters. We obtain N order rational solutions that can be written as a quotient of two polynomials of degree 2N(N+1) in x, t and 4N(N+1) in y depending on 2N-2 parameters. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x,y) plane in function of time t and parameters a1, a2, b1, and b2.http://dx.doi.org/10.1155/2018/1642139 |
| spellingShingle | Pierre Gaillard The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three Advances in Mathematical Physics |
| title | The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three |
| title_full | The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three |
| title_fullStr | The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three |
| title_full_unstemmed | The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three |
| title_short | The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three |
| title_sort | johnson equation fredholm and wronskian representations of solutions and the case of order three |
| url | http://dx.doi.org/10.1155/2018/1642139 |
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