Families of solutions to the KPI equation and the structure of their rational representations of order N

Pierre Gaillard (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

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. 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 until the order 6 and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters.

Mots clés: numbers : 33Q55; 37K10; 4710A-; 4735Fg; 4754Bd

Domaines: Physique mathématique [math-ph]

Pierre Gaillard. Families of solutions to the KPI equation and the structure of their rational representations of order N. 2018. ⟨hal-01865763⟩