8-parameter solutions of fifth order to the Johnson equation

Pierre Gaillard (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We give different representations of the solutions of the Johnson equation with parameters. First, an expression in terms of Fredholm determinants is given; we give also a representation of the solutions written as a quotient of wronskians of order 2N. These 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 polyno-mials of degree 2N (N +1) in x, t and 4N (N +1) in y depending on 2N −2 parameters. Here, we explicitly construct the expressions of the rational solutions of order 5 depending on 8 real parameters and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters ai and bi for 1 ≤ i ≤ 4.

Mots clés: rational solutions; Johnson equation; Fredholm determinants; wronskians; ratio- nal solutions; rogue waves PACS numbers : 33Q55; 37K10; 4710A-; 4735Fg; 4754Bd

Domaines: Physique mathématique [math-ph]

Pierre Gaillard. 8-parameter solutions of fifth order to the Johnson equation. 2019. ⟨hal-02268910⟩