This article is cited in 12 papers

Multiparametric families of solutions of the Kadomtsev–Petviashvili-I equation, the structure of their rational representations, and multi-rogue waves

P. Gaillard

Université de Bourgogne, Institut de mathématiques de Bourgogne, Faculté des Sciences Mirande, Dijon, France

Abstract: We construct solutions of the Kadomtsev–Petviashvili-I equation 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 up to the order six and study the patterns of their modulus in the plane $(x,y)$ and their evolution according to time and parameters.

Keywords: Kadomtsev–Petviashvili equation, Fredholm determinant, Wronskian, lump, rogue wave.

PACS: 33Q55, 37K10, 47.10A-, 47.35.Fg, 47.54.Bd

Received: 24.07.2017

DOI: 10.4213/tmf9435

 English version: Theoretical and Mathematical Physics, 2018, 196:2, 1174–1199

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