Solutions to the Gardner equation with multiparameters and the rational case

Pierre Gaillard (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

We construct solutions to the Gardner equation in terms of trigonometric and hyperbolic functions, depending on several real parameters. Using a passage to the limit when one of these parameters goes to 0, we get, for each positive integer N , rational solutions as a quotient of polynomials in x and t depending on 2N parameters. We construct explicit expressions of these rational solutions for orders N = 1 until N = 3. We easily deduce solutions to the mKdV equation in terms of wronskians as well as rational solutions depending on 2N real parameters.

Mots clés: Gardner equation; wronskians; rational solutions PACS numbers : 33Q55; 37K10; 47.10A-; 47.35.Fg; 47.54.Bd

Domaines: Physique mathématique [math-ph]

Pierre Gaillard. Solutions to the Gardner equation with multiparameters and the rational case. 2022. ⟨hal-03540008⟩