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.