Geodesic
Multiparametric families of solutions of the~Kadomtsev--Petviashvili-I equation, the~structure of their rational representations, and multi-rogue waves
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 266-293

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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
Mots-clés : rogue wave. 
@article{TMF_2018_196_2_a5,
     author = {P. Gaillard},
     title = {Multiparametric families of solutions of {the~Kadomtsev--Petviashvili-I} equation, the~structure of their rational representations, and multi-rogue waves},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {266--293},
     publisher = {mathdoc},
     volume = {196},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a5/}
}
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P. Gaillard. Multiparametric families of solutions of the~Kadomtsev--Petviashvili-I equation, the~structure of their rational representations, and multi-rogue waves. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 2, pp. 266-293. http://geodesic.mathdoc.fr/item/TMF_2018_196_2_a5/