# Multiparametric Families of Solutions of the Kadomtsev–Petviashvili-I Equation, the Structure of Their Rational Representations, and Multi-Rogue Waves

Article

First Online:

- 20 Downloads

## 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## Preview

Unable to display preview. Download preview PDF.

## References

- 1.B. B. Kadomtsev and V. I. Petviashvili, “On the stability of solitary waves in weakly dispersing media,” Sov. Phys. Dokl., 15, 539–541 (1970).Google Scholar
- 2.M. J. Ablowitz and H. Segur, “On the evolution of packets of water waves,” J. Fluid Mech.,
**92**, 691–715 (1979).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 3.D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. S. Kivshar, “Self–focusing of plane dark solitons in nonlinear defocusing media,” Phys. Rev. E,
**51**, 5016–5026 (1995).ADSMathSciNetCrossRefGoogle Scholar - 4.V. S. Dryuma, “Analytic solution of the two–dimensional Korteweg–de Vries (KdV) equation,” Lett. JETP,
**19**, 387–388 (1973).ADSGoogle Scholar - 5.S. V. Manakov, V. E. Zakharov, L. A. Bordag, and V. B. Matveev, “Two–dimensional solitons of the Kadomtsev–Petviashvili equation and their interaction,” Phys. Lett.,
**63**, 205–206 (1977).CrossRefGoogle Scholar - 6.I. M. Krichever, “Rational solutions of the Kadomtsev–Petviashvili equation and integrable systems of N particles on a line,” Funct. Anal. Appl.,
**12**, 59–61 (1978).CrossRefzbMATHGoogle Scholar - 7.I. M. Krichever and S. P. Novikov, “Holomorphic bundles over Riemann surfaces and the Kadomtsev–Petviashvili equation: I,” Funct. Anal. Appl.,
**12**, 276–286 (1978).CrossRefzbMATHGoogle Scholar - 8.B. A. Dubrovin, “Theta functions and non–linear equations,” Russ. Math. Surveys,
**36**, 11–92 (1981).ADSCrossRefzbMATHGoogle Scholar - 9.I. M. Krichever, “Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles,” Funct. Anal. Appl.,
**14**, 282–290 (1980).CrossRefzbMATHGoogle Scholar - 10.J. Satsuma and M. J. Ablowitz, “Two–dimensional lumps in nonlinear dispersive systems,” J. Math. Phys.,
**20**, 1496–1503 (1979).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 11.V. B. Matveev, “Darboux transformation and explicit solutions of the Kadomtcev–Petviaschvily equation depending on functional parameters,” Lett. Math. Phys.,
**3**, 213–216 (1979).ADSMathSciNetCrossRefGoogle Scholar - 12.J. J. C. Nimmo and N. C. Freeman, “Rational solutions of the KdV equation in wronskian form,” Phys. Lett. A,
**96**, 443–446 (1983).ADSMathSciNetCrossRefGoogle Scholar - 13.J. J. C. Nimmo and N. C. Freeman, “The use of Bäcklund transformations in obtaining N–soliton solutions in Wronskian form,” J. Phys. A: Math. Gen.,
**17**, 1415–1424 (1984).ADSCrossRefzbMATHGoogle Scholar - 14.V. B. Matveev and M. A. Salle, “New families of the explicit solutions of the Kadomtcev–Petviaschvily equation and their application to Johnson equation,” in:
*Some Topics on Inverse Problems (Montpellier, France, 30 November–4 December 1987, P. C. Sabatier, ed.), World Scientific, Singapore*(1987), pp. 304–315.Google Scholar - 15.D. E. Pelinovsky and Y. A. Stepanyants, “New multisolitons of the Kadomtsev–Petviashvili equation,” Phys. JETP Lett.,
**57**, 24–28 (1993).ADSGoogle Scholar - 16.D. E. Pelinovsky, “Rational solutions of the Kadomtsev–Petviashvili hierarchy and the dynamics of their poles: I. New form of a general rational solution,” J. Math. Phys.,
**35**, 5820–5830 (1994).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 17.M. J. Ablowitz and J. Villarroel, “Solutions to the time dependent Schrödinger and the Kadomtsev–Petviashvili equations,” Phys. Rev. Lett.,
**78**, 570–573 (1997).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 18.J. Villarroel and M. J. Ablowitz, “On the discrete spectrum of the nonstationary Schrödinger equation and multipole lumps of the Kadomtsev–Petviashvili I equation,” Commun. Math. Phys.,
**207**, 1–42 (1999).ADSCrossRefzbMATHGoogle Scholar - 19.M. J. Ablowitz, S. Chakravarty, A. D. Trubatch, and J. Villaroel, “A novel class of solution of the non–stationary Schrödinger and the KP equations,” Phys. Lett. A,
**267**, 132–146 (2000).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 20.G. Biondini and Y. Kodama, “On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy,” J. Phys. A: Math. Gen.,
**36**, 10519–10536 (2003).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 21.Y. Kodama, “Young diagrams and N solitons solutions to the KP equation,” J. Phys. A: Math. Gen.,
**37**, 11169–11190 (2004).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 22.G. Biondini, “Line soliton interactions of the Kadomtsev–Petviashvili equation,” Phys. Rev. Lett.,
**99**, 064103 (2007).ADSCrossRefGoogle Scholar - 23.P. Gaillard, “Rational solutions to the KPI equation and multi rogue waves,” Ann. Phys.,
**367**, 1–5 (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar - 24.P. Gaillard, “Fredholm and Wronskian representations of solutions to the KPI equation and multi–rogue waves,” J. Math. Phys.,
**57**, 063505 (2016).ADSMathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

© Pleiades Publishing, Ltd. 2018