A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

Bilman, Deniz; Miller, Peter D.

doi:10.1002/cpa.21819

Cited by 146 publications

(173 citation statements)

References 40 publications

(199 reference statements)

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“…Meanwhile, the classical inverse scattering method on plane wave background can not involve the rogue waves and high order rogue waves directly; we need to perform a sensitive limiting procedure to obtain them. Recently, Bilman and Miller [8] proposed a new scheme called robust inverse scattering transform to deal with the spectral singularity with the aid of the Darboux transformation. Meanwhile, it is hopeful to be used in analyzing the long-time asymptotics for the rogue waves and high order rogue wave on the plane wave background.…”

Section: Conclusion and Discussionmentioning

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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Section: Conclusion and Discussionmentioning

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 now introduce the basic Riemann-Hilbert problem used to define the multipole solitons we study. This representation was derived in [1] using the recently introduced robust inverse-scattering transform [3].…”

Section: Introductionmentioning

confidence: 99%

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

2019

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The integrable focusing nonlinear Schrödinger equation admits soliton solutions whose associated spectral data consist of a single pair of conjugate poles of arbitrary order. We study families of such multiple-pole solitons generated by Darboux transformations as the pole order tends to infinity. We show that in an appropriate scaling, there are four regions in the spacetime plane where solutions display qualitatively distinct behaviors: an exponential-decay region, an algebraic-decay region, a non-oscillatory region, and an oscillatory region. Using the nonlinear steepest-descent method for analyzing Riemann-Hilbert problems, we compute the leading-order asymptotic behavior in the algebraic-decay, non-oscillatory, and oscillatory regions.

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“…The finite-gap method was employed to relate the unstable modes on the constant background with the occurrence of rogue waves [24,25]. 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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A Robust Inverse Scattering Transform for the Focusing Nonlinear Schrödinger Equation

Cited by 146 publications

References 40 publications

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

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

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

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

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