Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case
From elementary exponential functions which depend on several parameters, we construct multi-parametric solutions to the Boussinesq equation. When we perform a passage to the limit when one of these para\-meters goes to $0$, we get rational solutions as a quotient of a polynomial of degree $N(N+1)-2...
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| Format: | Article |
| Language: | English |
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Emrah Evren KARA
2020-06-01
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| Series: | Universal Journal of Mathematics and Applications |
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| Online Access: | https://dergipark.org.tr/tr/download/article-file/1162320 |
| _version_ | 1797350206059053056 |
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| author | Pierre Gaillard |
| author_facet | Pierre Gaillard |
| author_sort | Pierre Gaillard |
| collection | DOAJ |
| description | From elementary exponential functions which depend on several parameters, we construct multi-parametric solutions to the Boussinesq equation. When we perform a passage to the limit when one of these para\-meters goes to $0$, we get rational solutions as a quotient of a polynomial of degree $N(N+1)-2$ in $x$ and $t$, by a polynomial of degree $N(N+1)$ in $x$ and $t$ for each positive integer $N$ depending on $3N$ real parameters. We restrict ourself to give the explicit expressions of these rational solutions for $N=1$ until $N=3$ to shortened the paper. We easily deduce the corresponding explicit rational solutions to the Kadomtsev Petviashvili equation for the same orders from $1$ to $3$. |
| first_indexed | 2024-03-08T12:41:20Z |
| format | Article |
| id | doaj.art-72f3ca3ed6184f1d9031e9ad66d3eec5 |
| institution | Directory Open Access Journal |
| issn | 2619-9653 |
| language | English |
| last_indexed | 2024-03-08T12:41:20Z |
| publishDate | 2020-06-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Universal Journal of Mathematics and Applications |
| spelling | doaj.art-72f3ca3ed6184f1d9031e9ad66d3eec52024-01-21T09:52:01ZengEmrah Evren KARAUniversal Journal of Mathematics and Applications2619-96532020-06-0132445210.32323/ujma.6448371225Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational CasePierre Gaillard0Université de BourgogneFrom elementary exponential functions which depend on several parameters, we construct multi-parametric solutions to the Boussinesq equation. When we perform a passage to the limit when one of these para\-meters goes to $0$, we get rational solutions as a quotient of a polynomial of degree $N(N+1)-2$ in $x$ and $t$, by a polynomial of degree $N(N+1)$ in $x$ and $t$ for each positive integer $N$ depending on $3N$ real parameters. We restrict ourself to give the explicit expressions of these rational solutions for $N=1$ until $N=3$ to shortened the paper. We easily deduce the corresponding explicit rational solutions to the Kadomtsev Petviashvili equation for the same orders from $1$ to $3$.https://dergipark.org.tr/tr/download/article-file/1162320boussinesq equationdeterminantslax pairsrational solutions |
| spellingShingle | Pierre Gaillard Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case Universal Journal of Mathematics and Applications boussinesq equation determinants lax pairs rational solutions |
| title | Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case |
| title_full | Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case |
| title_fullStr | Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case |
| title_full_unstemmed | Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case |
| title_short | Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case |
| title_sort | multi parametric families of solutions of order n to the boussinesq and kp equations and the degenerate rational case |
| topic | boussinesq equation determinants lax pairs rational solutions |
| url | https://dergipark.org.tr/tr/download/article-file/1162320 |
| work_keys_str_mv | AT pierregaillard multiparametricfamiliesofsolutionsoforderntotheboussinesqandkpequationsandthedegeneraterationalcase |