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

Authors : Pierre GAİLLARD

Pages : 44-52

Doi:10.32323/ujma.644837

View : 15 | Download : 11

Publication Date : 2020-06-22

Article Type : Research Paper

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 $Ninsert ignore into journalissuearticles values(N+1);-2$ in $x$ and $t$, by a polynomial of degree $Ninsert ignore into journalissuearticles values(N+1);$ 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