Multiparametric families of solutions of the Kadomtsev–Petviashvili-I equation, the structure of their rational representations, and multi-rogue waves

We construct solutions of the Kadomtsev–Petviashvili-I equation in terms of Fredholm determinants. We deduce solutions written as a quotient of Wronskians of order $2N$. These solutions, called solutions of order $N$, depend on $2N{-}1$ parameters. They can also be written as a quotient of two polynomials of degree $2N(N+1)$ in $x$, $y$, and $t$ depending on $2N-2$ parameters. The maximum of the modulus of these solutions at order $N$ is equal to $2(2N+1)^2$. We explicitly construct the expressions up to the order six and study the patterns of their modulus in the plane $(x,y)$ and their evolution according to time and parameters.