Solutions to the NLS equation : differential relations and their different representations

Pierre Gaillard (1)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]

Solutions to the focusing nonlinear Schrödinger 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 quasirational solutions to the NLS equation denoted by vN and have been explicitly constructed until order N = 13. These solutions appear as deformations of the Peregrine breather PN 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.

Mots clés: Fredholm determinants; NLS equation; Peregrine breathers; rogue waves; wronskians

Domaines: Physique mathématique [math-ph]

Pierre Gaillard. Solutions to the NLS equation : differential relations and their different representations. 2020. ⟨hal-03091896⟩