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.