https://hal-univ-bourgogne.archives-ouvertes.fr/hal-02521648Gaillard, PierrePierreGaillardIMB - Institut de Mathématiques de Bourgogne [Dijon] - UB - Université de Bourgogne - UBFC - Université Bourgogne Franche-Comté [COMUE] - CNRS - Centre National de la Recherche ScientifiqueDifferential Relations for the Solutions to the NLS Equation and Their Different RepresentationsHAL CCSD2019Fredholm determinantsNLS equationPeregrine breathersRogue wavesWronskians[MATH] Mathematics [math]université de Bourgogne, IMB -2020-03-27 15:07:552022-12-11 09:11:382020-03-27 15:07:55enJournal articles10.33434/cams.5580441Solutions to the focusing nonlinear Schr ̈odinger equation (NLS) of orderNdepending 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 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.