Other Families of Rational Solutions to the KPI Equation

International audienceAims / Objectives: We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in $x$, $y$ and $t$ depending on several real parameters. We get an infinite hierarchy of rational solutions written as a quotient of a polynomial of degree $2N(N + 1) - 2$ in $x$, $y$ and $t$ by a polynomial of degree $2N(N + 1)$ in $x$, $y$ and $t$, depending on $2N - 2$ real parameters for each positive integer $N$. Place and Duration of Study: Institut de mathématiques de Bourgogne, Université de Bourgogne Franche-Conté between January 2020 and January 2021. Conclusion: We construct 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. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of two peaks in the case of order 2