Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy
This study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are g...
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MDPI AG
2024-11-01
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| Series: | AppliedMath |
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| author | Pierre Gaillard |
| author_facet | Pierre Gaillard |
| author_sort | Pierre Gaillard |
| collection | DOAJ |
| description | This study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are given for the first order. These solutions depend on multiple real parameters. We study the associated patterns of these solutions in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> plane according to the different values of their parameters. This work allows us to highlight the phenomenon of rogue waves, such as those seen in the case of lower-order equations such as the nonlinear Schrödinger equation, the mKdV equation, or the Hirota equation. |
| format | Article |
| id | doaj-art-fe6fac8c01ae4401a8d0343c7ff7940c |
| institution | Kabale University |
| issn | 2673-9909 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | AppliedMath |
| spelling | doaj-art-fe6fac8c01ae4401a8d0343c7ff7940c2024-12-27T14:07:09ZengMDPI AGAppliedMath2673-99092024-11-01441418142710.3390/appliedmath4040075Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger HierarchyPierre Gaillard0Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche Comté, 9 Avenue Alain Savary, BP 47870, 21078 Dijon Cedex, FranceThis study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are given for the first order. These solutions depend on multiple real parameters. We study the associated patterns of these solutions in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula> plane according to the different values of their parameters. This work allows us to highlight the phenomenon of rogue waves, such as those seen in the case of lower-order equations such as the nonlinear Schrödinger equation, the mKdV equation, or the Hirota equation.https://www.mdpi.com/2673-9909/4/4/75NLS hierarchyrational solutionsNLS equationmKdV equationHirota equationrogue waves |
| spellingShingle | Pierre Gaillard Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy AppliedMath NLS hierarchy rational solutions NLS equation mKdV equation Hirota equation rogue waves |
| title | Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy |
| title_full | Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy |
| title_fullStr | Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy |
| title_full_unstemmed | Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy |
| title_short | Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy |
| title_sort | rational solutions to the fourth equation of the nonlinear schrodinger hierarchy |
| topic | NLS hierarchy rational solutions NLS equation mKdV equation Hirota equation rogue waves |
| url | https://www.mdpi.com/2673-9909/4/4/75 |
| work_keys_str_mv | AT pierregaillard rationalsolutionstothefourthequationofthenonlinearschrodingerhierarchy |