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

Abstract : Solutions 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.
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Soumis le : vendredi 27 mars 2020 - 15:07:55
Dernière modification le : vendredi 25 novembre 2022 - 10:12:06

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Pierre Gaillard. Differential Relations for the Solutions to the NLS Equation and Their Different Representations. Communications in Advanced Mathematical Sciences, 2019, II (4), pp.235-243. ⟨10.33434/cams.558044⟩. ⟨hal-02521648⟩



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