Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation

Wright, Otis C.

doi:10.1016/j.physd.2016.03.002

Cited by 22 publications

(21 citation statements)

References 24 publications

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“…These representations coincide with those used in [41] for integration of ultra-elliptic solutions to the cubic NLS equation.…”

Section: Degeneration Proceduressupporting

confidence: 62%

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

Chen¹,

Pelinovsky²

2019

J Nonlinear Sci

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We address the most general periodic travelling wave of the modified Korteweg-de Vries (mKdV) equation written as a rational function of Jacobian elliptic functions. By applying an algebraic method which relates the periodic travelling waves and the squared periodic eigenfunctions of the Lax operators, we characterize explicitly the location of eigenvalues in the periodic spectral problem away from the imaginary axis. We show that Darboux transformations with the periodic eigenfunctions remain in the class of the same periodic travelling waves of the mKdV equation. In a general setting, there exist three symmetric pairs of simple eigenvalues away from the imaginary axis, and we give a new representation of the second non-periodic solution to the Lax equations for the same eigenvalues. We show that Darboux transformations with the non-periodic solutions to the Lax equations produce rogue waves on the periodic background, which are either brought from infinity by propagating algebraic solitons or formed in a finite region of the time-space plane.

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“…These representations coincide with those used in [41] for integration of ultra-elliptic solutions to the cubic NLS equation.…”

Section: Degeneration Proceduressupporting

confidence: 62%

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

Chen¹,

Pelinovsky²

2019

J Nonlinear Sci

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“…In fact, due to ergodic property of quasi-periodic solutions, this maximum will be approached by a finite-gap solution with a given spectral bands and generic initial phases in a sufficiently large space-time region. In the case of genus two, this result was recently obtained by O. Wright in [21]. It turns out that the obtained formula is also valid for finite-gap solutions of the defocusing NLS (dNLS) and that a somewhat similar statement is valid for KdV.…”

supporting

confidence: 60%

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

Bertola

Tovbis

2017

Commun. Math. Phys.

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In this paper we prove that the maximum amplitude of a finite-gap solution to the focusing Nonlinear Schrödinger equation with given spectral bands does not exceed half of the sum of the length of all the bands. This maximum will be attained for certain choices of the initial phases. A similar result is also true for the defocusing Nonlinear Schrödinger equation. 1 arXiv:1601.00875v1 [math-ph] 5 Jan 2016 phenomena. Here we refer to the common understanding of rogue waves as exceptionally tall waves with the amplitude |ψ| 2 ≥ 8|ψ| 2 0 , where |ψ 0 | is the amplitude of the background waves. Several particular types of solutions to the fNLS expressed through elementary functions (Peregrine, Akhmediev and Kuznetsov-Ma breathers, see, for example, [10]) provide, perhaps, the most known examples of the rogue wave solutions. These breathers can be viewed as degenerate limits of the corresponding finite-gap solutions.Therefore, it appears natural to look for rogue waves in the class of finite-gap solutions to fNLS. This problem is the subject of an ongoing research [4] for finite-gap solutions of any genus (which is the number of the spectral bands minus one). The main goal of the present paper is a new simple formula for the maximal amplitude of a finite-gap solution with given spectral bands. Namely, we proved that the maximal amplitude cannot exceed half of the sum of the length of all the spectral bands, and this maximum will be attained for certain choices of the initial phases. In fact, due to ergodic property of quasi-periodic solutions, this maximum will be approached by a finite-gap solution with a given spectral bands and generic initial phases in a sufficiently large space-time region. In the case of genus two, this result was recently obtained by O. Wright in [21]. It turns out that the obtained formula is also valid for finite-gap solutions of the defocusing NLS (dNLS) and that a somewhat similar statement is valid for KdV. It will be convenient to describe the finite-gap solutions through the corresponding Riemann-Hilbert Problems (RHPs). It is well known that the inverse scattering transform (IST) method of solving nonlinear integrable systems can be reduced to certain matrix RHP (see [17], [16], [22]), where the jump matrices are defined in terms of the scattering data. The RHPs with permutation type piece-wise constant jump matrices correspond to finite-gap solutions ([6], [5]). In the context of the semiclassical (small dispersion limit) analysis, such RHPs (known as model RHPs or outer parametrices) were first studied in [7] for the KdV and in [18], [15] for the fNLS. They represent the leading order term of the original RHP. Model RHPs are usually obtained through the nonlinear steepest descent method of Deift and Zhou. The finite-gap solution of a model problem provides the local (in x, t) leading order behavior (in the semiclassical limit) of the corresponding slowly modulated solution.Description of results. The data that characterize a finite-gap solution is: (a) a hyperelliptic Riemann surface R...

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“…was found in [8] using the g-band solution in the form (2.3) (for g = 2 case this result was established in [38]; its particular case for SFBs appears in [28] 1 ). Importantly, if the wavenumbers k j are incommensurable, the value |ψg|m is the supremum of |ψg| over x ∈ R for any fixed t.…”

Section: Rogue Waves On a Finite-band Potential Backgroundmentioning

confidence: 84%

“…Given some formal amplitude criterion for such a rogue wave event, it is clear that not all finite-band solutions of the fNLS equation can exhibit rogue waves. By employing the recently obtained [8] explicit formula for the maximum of the wave field amplitude for finite-band potentials, see (2.9), (for the genus 2 case this result was established in [38]), we provide a simple analytical criterion for distinguishing the finite-band potentials exhibiting (generalised) rogue waves.…”

Section: Introductionmentioning

confidence: 95%

Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

Bertola

Tovbis

2016

Proc. R. Soc. A.

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Rogue waves appearing on deep water or in optical fibres are often modelled by certain breather solutions of the focusing nonlinear Schrödinger (fNLS) equation which are referred to as solitons on finite background (SFBs). A more general modelling of rogue waves can be achieved via the consideration of multiphase, or finite-band, fNLS solutions of whom the standard SFBs and the structures forming due to their collisions represent particular, degenerate, cases. A generalized rogue wave notion then naturally enters as a large-amplitude localized coherent structure occurring within a finite-band fNLS solution. In this paper, we use the winding of real tori to show the mechanism of the appearance of such generalized rogue waves and derive an analytical criterion distinguishing finite-band potentials of the fNLS equation that exhibit generalized rogue waves.

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Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation

Cited by 22 publications

References 24 publications

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

Periodic Travelling Waves of the Modified KdV Equation and Rogue Waves on the Periodic Background

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

Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation

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