Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

Bilman, Deniz; Buckingham, Robert

doi:10.1007/s00332-019-09542-7

Cited by 75 publications

(108 citation statements)

References 41 publications

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“…Rogue waves of infinite order were constructed in [10] based on recent developments in the inverse scattering method [9]. Rogue waves of the soliton superposition were studied asymptotically in the limit of many solitons [8,30].…”

Section: Introductionmentioning

confidence: 99%

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

2019

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The double-periodic solutions of the focusing nonlinear Schrödinger equation have been previously obtained by the method of separation of variables. We construct these solutions by using an algebraic method with two eigenvalues. Furthermore, we characterize the Lax spectrum for the double-periodic solutions and analyze rogue waves arising on their background. Magnification of the rogue waves is studied numerically.

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Section: Introductionmentioning

confidence: 99%

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

2019

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“…Rogue waves in physics are associated with the gigantic waves on the ocean's surface and in optical fibers which arise due to the modulation instability of the background wave [29,30]. Several recent publications were devoted to numerical and analytical studies of rogue waves on the background of periodic [3,23], quasi-periodic [5,6,12], and multi-soliton [7,8,9] wave patterns in the framework of the nonlinear Schrödinger (NLS) equation. Since internal waves are modeled by the mKdV equation [22], formation of rogue internal waves was also studied in the framework of the mKdV equation (1.1) as a result of multi-soliton interactions [34,35,36,37].…”

Section: Introductionmentioning

confidence: 99%

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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“…The first method is to take the limit in the higher-genus solutions constructed by the algebro-geometric method [3] or the Riemann-Hilbert method. The other way is using the inverse scattering method or the Darboux transformation [7,9,15,20,21,30] to construct the breathers by the linear algebra. However, both methods have different Date: December 12, 2018. difficulties.…”

Section: Introductionmentioning

confidence: 99%

Multi‐breather and high‐order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

2019

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We construct the multi-breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit t → ±∞, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi-and higher-order rogue wave solution.

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Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

Cited by 75 publications

References 41 publications

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

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

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

Multi‐breather and high‐order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

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