Finite-gap (algebro-geometric) solutions to the focusing Nonlinear SchrÃ¶dinger Equation (fNLS)(Formula presented.) are quasi-periodic solutions that represent nonlinear multi-phase waves. In general, a finite-gap solution for (0-1) is defined by a collection of Schwarz symmetrical spectral bands and of real constants (initial phases), associated with the corresponding bands. In this paper we prove an interesting new formula for the maximal amplitude of a finite-gap solution to the focusing Nonlinear SchrÃ¶dinger equation with given spectral bands: the amplitude does not exceed the sum of the imaginary parts of all the endpoints in the upper half plane. In the case of the straight vertical bands, that amounts to the half of the sum of the length of all the bands. The maximal amplitude will be attained for certain choices of the initial phases. This result is an important part of a criterion for the potential presence of the rogue waves in finite-gap solutions with a given set of spectral endpoints, obtained in Bertola etÂ al. (Proc R Soc A, 2016. doi:10.1098/rspa.2016.0340). A similar result was also obtained for the defocusing Nonlinear SchrÃ¶dinger equation.

Maximal amplitudes of finite-gap solutions for the focusing Nonlinear Schrödinger Equation / Bertola, M.; Tovbis, A.. - In: COMMUNICATIONS IN MATHEMATICAL PHYSICS. - ISSN 0010-3616. - 354:2(2017), pp. 525-547. [10.1007/s00220-017-2895-9]

### Maximal amplitudes of finite-gap solutions for the focusing Nonlinear Schrödinger Equation

#### Abstract

Finite-gap (algebro-geometric) solutions to the focusing Nonlinear SchrÃ¶dinger Equation (fNLS)(Formula presented.) are quasi-periodic solutions that represent nonlinear multi-phase waves. In general, a finite-gap solution for (0-1) is defined by a collection of Schwarz symmetrical spectral bands and of real constants (initial phases), associated with the corresponding bands. In this paper we prove an interesting new formula for the maximal amplitude of a finite-gap solution to the focusing Nonlinear SchrÃ¶dinger equation with given spectral bands: the amplitude does not exceed the sum of the imaginary parts of all the endpoints in the upper half plane. In the case of the straight vertical bands, that amounts to the half of the sum of the length of all the bands. The maximal amplitude will be attained for certain choices of the initial phases. This result is an important part of a criterion for the potential presence of the rogue waves in finite-gap solutions with a given set of spectral endpoints, obtained in Bertola etÂ al. (Proc R Soc A, 2016. doi:10.1098/rspa.2016.0340). A similar result was also obtained for the defocusing Nonlinear SchrÃ¶dinger equation.
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2017
354
2
525
547
https://arxiv.org/abs/1601.00875
Bertola, M.; Tovbis, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/20.500.11767/68404`