Rational solutions to the KPI equation and multi rogue waves
Résumé
We construct here rational solutions to the Kadomtsev-Petviashvili equation (KPI) as a quotient of two polynomials in $x, y$ and $t$ depending on several real parameters. This method provides an infinite hierarchy of rational solutions written in terms of polynomials of degrees $2N (N + 1)$ in $x, y$ and $t$ depending on $2N - 2$ real parameters for each positive integer $N$. We give explicit expressions of the solutions in the simplest cases $N = 1$ and $N = 2$ and we study the patterns of their modulus in the $(x, y)$ plane for different values of time t and parameters.