Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

Chen, Jinbing; Pelinovsky, Dmitry E.; White, Rob

doi:10.1103/physreve.100.052219

Cited by 94 publications

(86 citation statements)

References 42 publications

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“…Differing from these, very recently, Chen and his collaborators have reported an algorithm to construct RW solutions on the periodic wave background for the focusing NLS equation [31]. The same authors have also derived RW solutions on the double-periodic background for the focusing NLS equation using an effective algebraic method [32]. Subsequently, Peng et al have studied the characteristics of RWs on the periodic background for the Hirota equation [33].…”

Section: Introductionmentioning

confidence: 99%

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Rogue waves on the double-periodic background in Hirota equation

2021

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We construct rogue wave solutions on the double-periodic background for the Hirota equation through onefold Darboux transformation formula. We consider two types of double-periodic solutions as seed solutions. We identify the squared eigenfunctions and eigenvalues that appear in the onefold Darboux transformation formula through an algebraic method with two eigenvalues. We then construct the desired solution in two steps. In the first step, we create double-periodic waves as the background. In the second step, we build rogue wave solution on the top of this double-periodic solution. We present the localized structures for two different values of the arbitrary parameter each one with two different sets of eigenvalues.

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Section: Introductionmentioning

confidence: 99%

“…Since the spatial part is same, we obtain the same sets of differential constraints which connect the potential with eigenvalues that are given in Ref. [32]. We construct periodic eigenfunctions by solving the differential constraints in terms of known double-periodic solutions of the Hirota equation.…”

Section: Introductionmentioning

confidence: 99%

Rogue waves on the double-periodic background in Hirota equation

2021

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“…In the case of the Stokes waves with N unstable modes (UMs), the associated SPBs can be of dimension M ≤ N and are referred to as M-mode SPBs; the single mode SPB is the Akhmediev breather [13]. For more realistic sea states with non-uniform backgrounds, heteroclinic orbits of unstable N-phase solutions have been used to describe rogue waves [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation

Schober

Islas

2021

Front. Phys.

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Spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation are frequently used to model rogue waves and are typically unstable. In this paper we study the effects of dissipation and higher order nonlinearities on the stabilization of N-mode SPBs, 1≤N≤3, in the framework of a damped higher order NLS (HONLS) equation. We observe the onset of novel instabilities associated with the development of critical states resulting from symmetry breaking in the damped HONLS system. We develop a broadened Floquet characterization of instabilities of solutions of the NLS equation by showing that instabilities are associated with degenerate complex elements of not only the discrete, but also the continuous Floquet spectrum. As a result, the Floquet criteria for the stabilization of a solution of the damped HONLS centers around the elimination of all complex degenerate elements of the spectrum. For a given initial N-mode SPB, a short-time perturbation analysis shows that the complex double points associated with resonant modes split under the damped HONLS while those associated with nonresonant modes remain closed. The corresponding /damped HONLS numerical experiments corroborate that instabilities associated with nonresonant modes persist on a longer time scale than the instabilities associated with resonant modes.

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“…Recently, there has also been an intense research on the socalled periodic Peregrine soliton, by which we mean a Peregrine soliton formed on a periodic background [35][36][37][38][39][40][41]. Normally, when a multicomponent nonlinear system is confronted, it may occur to us that an interference would occur when two or more monochromatic waves of different frequencies are simultaneously present in the same region.…”

Section: Introductionmentioning

confidence: 99%

Peregrine Solitons on a Periodic Background in the Vector Cubic-Quintic Nonlinear Schrödinger Equation

Wang

et al. 2020

Front. Phys.

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Rogue waves on the double-periodic background in the focusing nonlinear Schrödinger equation

Cited by 94 publications

References 42 publications

Rogue waves on the double-periodic background in Hirota equation

Rogue waves on the double-periodic background in Hirota equation

On the Stabilization of Breather-type Solutions of the Damped Higher Order Nonlinear Schrödinger Equation

Peregrine Solitons on a Periodic Background in the Vector Cubic-Quintic Nonlinear Schrödinger Equation

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