Multi-Parametric Families of Solutions of Order $N$ to the Boussinesq and KP Equations and the Degenerate Rational Case
Résumé
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 parameters goes to0, we get rational solutions as a quotient of apolynomial of degree N(N+1)−2inx and Boussinesq equation,KadomtsevPetviashviliequation,Rational solutionst, 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 fo rN=1 until N=3to shortened thepaper. We easily deduce the corresponding explicit rational solutions to the Kadomtsev Petviashvili equation for the same orders from 1 to 3.