From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N − 2 parameters, the structure of the solutions and the case of fourth order

Pierre Gaillard (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We have already constructed solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants and wronskians of order 2N. These solutions have been called solutions of order N and they depend on 2N − 1 parameters. We construct here N-order rational solutions. We prove that they can be written as a quotient of 2 polynomials of degree 2N (N + 1) in x, y and t depending on 2N − 2 parameters. We explicitly construct the expressions of the rational solutions of order 4 depending on 6 real parameters and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, b1, b2, b3.

Domaines: Physique mathématique [math-ph]

Pierre Gaillard. From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N − 2 parameters, the structure of the solutions and the case of fourth order. 2017. ⟨hal-01525384v1⟩