Differential Relations for the Solutions to the NLS Equation and Their Different Representations

Pierre Gaillard

doi:10.33434/cams.558044

EN

Differential Relations for the Solutions to the NLS Equation and Their Different Representations

Abstract

Solutions to the focusing nonlinear Schr\"odinger equation (NLS) of order $N$ depending on $2N-2$ real parameters in terms of wronskians and Fredholm determinants are given. These solutions give families of quasi-rational solutions to the NLS equation denoted by $v_{N}$ and have been explicitly constructed until order $N = 13$. These solutions appear as deformations of the Peregrine breather $P_{N}$ as they can be obtained when all parameters are equal to $0$. These quasi rational solutions can be expressed as a quotient of two polynomials of degree $N(N+1)$ in the variables $x$ and $t$ and the maximum of the modulus of the Peregrine breather of order $N$ is equal to $2N+1$. \\ Here we give some relations between solutions to this equation. In particular, we present a connection between the modulus of these solutions and the denominator part of their rational expressions. Some relations between numerator and denominator of the Peregrine breather are presented.

Keywords

NLS equation,Fredholm determinants,NLS equation,Peregrine breathers,Rogue waves,Wronskians,Wronskians

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Pierre Gaillard ^*
0000-0002-7073-8284
France

Publication Date

December 29, 2019

Submission Date

April 25, 2019

Acceptance Date

August 6, 2019

Published in Issue

Year 2019 Volume: 2 Number: 4

DOI

https://doi.org/10.33434/cams.558044

IZ

https://izlik.org/JA78YB83AN

APA

Gaillard, P. (2019). Differential Relations for the Solutions to the NLS Equation and Their Different Representations. Communications in Advanced Mathematical Sciences, 2(4), 235-243. https://doi.org/10.33434/cams.558044

AMA

1.Gaillard P. Differential Relations for the Solutions to the NLS Equation and Their Different Representations. Communications in Advanced Mathematical Sciences. 2019;2(4):235-243. doi:10.33434/cams.558044

Chicago

Gaillard, Pierre. 2019. “Differential Relations for the Solutions to the NLS Equation and Their Different Representations”. Communications in Advanced Mathematical Sciences 2 (4): 235-43. https://doi.org/10.33434/cams.558044.

EndNote

Gaillard P (December 1, 2019) Differential Relations for the Solutions to the NLS Equation and Their Different Representations. Communications in Advanced Mathematical Sciences 2 4 235–243.

IEEE

[1]P. Gaillard, “Differential Relations for the Solutions to the NLS Equation and Their Different Representations”, Communications in Advanced Mathematical Sciences, vol. 2, no. 4, pp. 235–243, Dec. 2019, doi: 10.33434/cams.558044.

ISNAD

Gaillard, Pierre. “Differential Relations for the Solutions to the NLS Equation and Their Different Representations”. Communications in Advanced Mathematical Sciences 2/4 (December 1, 2019): 235-243. https://doi.org/10.33434/cams.558044.

JAMA

1.Gaillard P. Differential Relations for the Solutions to the NLS Equation and Their Different Representations. Communications in Advanced Mathematical Sciences. 2019;2:235–243.

MLA

Gaillard, Pierre. “Differential Relations for the Solutions to the NLS Equation and Their Different Representations”. Communications in Advanced Mathematical Sciences, vol. 2, no. 4, Dec. 2019, pp. 235-43, doi:10.33434/cams.558044.

Vancouver

1.Pierre Gaillard. Differential Relations for the Solutions to the NLS Equation and Their Different Representations. Communications in Advanced Mathematical Sciences. 2019 Dec. 1;2(4):235-43. doi:10.33434/cams.558044

The published articles in CAMS are licensed under a Creative Commons Attribution-NonCommercial 4.0 International License..

Abstract

Differential Relations for the Solutions to the NLS Equation and Their Different Representations

Abstract

Keywords

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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