TY  - JOUR
T1  - Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case
AU  - Gaillard, Pierre
PY  - 2020
DA  - June
Y2  - 2020
DO  - 10.32323/ujma.644837
JF  - Universal Journal of Mathematics and Applications
JO  - Univ. J. Math. Appl.
PB  - Emrah Evren KARA
WT  - DergiPark
SN  - 2619-9653
SP  - 44
EP  - 52
VL  - 3
IS  - 2
LA  - en
AB  - 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 $3$.
KW  - Boussinesq equation
KW  - determinants
KW  - Lax pairs
KW  - rational solutions
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UR  - https://doi.org/10.32323/ujma.644837
L1  - https://dergipark.org.tr/en/download/article-file/1162320
ER  -