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

Year 2019, Volume: 2 Issue: 4, 235 - 243, 29.12.2019

Pierre Gaillard

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

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

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