Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations
This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and...
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MDPI AG
2025-03-01
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| author | Petar Popivanov Angela Slavova |
| author_facet | Petar Popivanov Angela Slavova |
| author_sort | Petar Popivanov |
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| description | This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude <i>Q</i> multiplied by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>e</mi><mrow><mi>i</mi><mo>Φ</mo></mrow></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Φ</mo></semantics></math></inline-formula> being linear. Then, <i>Q</i> is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included. |
| format | Article |
| id | doaj-art-3608e85ff7d143049bbcb520077191ae |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-3608e85ff7d143049bbcb520077191ae2025-08-20T03:43:27ZengMDPI AGMathematics2227-73902025-03-0113690110.3390/math13060901Traveling-Wave Solutions of Several Nonlinear Mathematical Physics EquationsPetar Popivanov0Angela Slavova1Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, BulgariaInstitute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, BulgariaThis paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude <i>Q</i> multiplied by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>e</mi><mrow><mi>i</mi><mo>Φ</mo></mrow></msup></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Φ</mo></semantics></math></inline-formula> being linear. Then, <i>Q</i> is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included.https://www.mdpi.com/2227-7390/13/6/901nonlinear Schrödinger equationconcatenation model from nonlinear fiber opticsKadomtsev–Petviashvili equationsolitonsAiry functionAbel’s ODE of the second kind |
| spellingShingle | Petar Popivanov Angela Slavova Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations Mathematics nonlinear Schrödinger equation concatenation model from nonlinear fiber optics Kadomtsev–Petviashvili equation solitons Airy function Abel’s ODE of the second kind |
| title | Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations |
| title_full | Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations |
| title_fullStr | Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations |
| title_full_unstemmed | Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations |
| title_short | Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations |
| title_sort | traveling wave solutions of several nonlinear mathematical physics equations |
| topic | nonlinear Schrödinger equation concatenation model from nonlinear fiber optics Kadomtsev–Petviashvili equation solitons Airy function Abel’s ODE of the second kind |
| url | https://www.mdpi.com/2227-7390/13/6/901 |
| work_keys_str_mv | AT petarpopivanov travelingwavesolutionsofseveralnonlinearmathematicalphysicsequations AT angelaslavova travelingwavesolutionsofseveralnonlinearmathematicalphysicsequations |