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

Authors : Pierre GAİLLARD

Pages : 235-243

Doi:10.33434/cams.558044

View : 20 | Download : 11

Publication Date : 2019-12-29

Article Type : Research Paper

Abstract :Solutions to the focusing nonlinear Schr\`odinger equation insert ignore into journalissuearticles values(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 $Ninsert ignore into journalissuearticles values(N+1);$ 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