Fredholm and Wronskian representations of solutions to the KPI equation and multi-rogue waves

We construct solutions to the Kadomtsev-Petviashvili equation (KPI) 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. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polynomials of degree 2N( N + 1) in x, y, and t depending on 2N - 2 parameters. So we get with this method an infinite hierarchy of solutions to the KPI equation. Published by AIP Publishing.