Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case

Pierre Gaillard

doi:10.32323/ujma.644837

EN

Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case

Abstract

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 $3$.

Keywords

Boussinesq equation,determinants,Lax pairs,rational solutions

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Pierre Gaillard ^*
0000-0002-7073-8284
France

Publication Date

June 22, 2020

Submission Date

November 10, 2019

Acceptance Date

March 26, 2020

Published in Issue

Year 2020 Volume: 3 Number: 2

DOI

https://doi.org/10.32323/ujma.644837

IZ

https://izlik.org/JA89FC32WP

APA

Gaillard, P. (2020). 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, 3(2), 44-52. https://doi.org/10.32323/ujma.644837

AMA

1.Gaillard P. Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case. Univ. J. Math. Appl. 2020;3(2):44-52. doi:10.32323/ujma.644837

Chicago

Gaillard, Pierre. 2020. “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 3 (2): 44-52. https://doi.org/10.32323/ujma.644837.

EndNote

Gaillard P (June 1, 2020) 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 3 2 44–52.

IEEE

[1]P. Gaillard, “Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case”, Univ. J. Math. Appl., vol. 3, no. 2, pp. 44–52, June 2020, doi: 10.32323/ujma.644837.

ISNAD

Gaillard, Pierre. “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 3/2 (June 1, 2020): 44-52. https://doi.org/10.32323/ujma.644837.

JAMA

1.Gaillard P. Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case. Univ. J. Math. Appl. 2020;3:44–52.

MLA

Gaillard, Pierre. “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, vol. 3, no. 2, June 2020, pp. 44-52, doi:10.32323/ujma.644837.

Vancouver

1.Pierre Gaillard. Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case. Univ. J. Math. Appl. 2020 Jun. 1;3(2):44-52. doi:10.32323/ujma.644837