- Universal Journal of Mathematics and Applications
- Vol: 3 Issue: 2
- Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degen...
Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case
Authors : Pierre Gaillard
Pages : 44-52
Doi:10.32323/ujma.644837
View : 7 | Download : 3
Publication Date : 2020-06-22
Article Type : Research
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