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Dynamics of rogue waves on a multi-soliton background in a vector nonlinear Schrodinger equation

Gui Mu, Zhenyun Qin, Roger Grimshaw

arXiv:1404.2988v1nlin.SI

TL;DR

Higher-order rogue-wave solutions for the vector nonlinear Schrödinger equation are sought beyond the low-order forms already reported. The paper combines Darboux dressing with asymptotic expansion to construct Nth-order solutions. These solutions exhibit varied rogue-wave and multisoliton patterns, revealing rich dynamics in the vector system.

  • Problem

    General higher-order rogue-wave solutions for the vector nonlinear Schrödinger equation are sought beyond the previously reported first-, second-, and third-order forms.

    Source: §1 · p. 3

  • Method

    A Darboux-dressing transformation combined with an asymptotic expansion constructs Nth-order rogue-wave solutions of the Manakov system.

    Source: §6 · p. 17

  • Results

    The solutions exhibit bright-dark higher-order rogue waves, bright-dark multi-rogue-wave patterns, and rogue waves interacting with multisolitons.

    Source: §6 · pp. 17, 22

  • Takeaways & Limitations

    The spatial-temporal patterns reveal rich dynamics in rogue-wave solutions of a vector system rather than the usual single-component NLSE.

    Source: §6 · p. 22

    The solutions demonstrated in the paper set the wavenumber k = 0; the authors state that the procedure can also be used for k ≠ 0.

    Source: §6 · p. 22

Abstract

from arXiv · show

General higher order rogue waves of a vector nonlinear Schrodinger equation (Manakov system) are derived using a Darboux-dressing transformation with an asymptotic expansion method. The Nth order semi-rational solutions containing 3N free parameters are expressed in separation of variables form. These solutions exhibit rogue waves on a multisoliton background. They demonstrate that the structure of rogue waves in this two-component system is richer than that in a one-component system. The study of our results would be of much importance in understanding and predicting rogue wave phenomena arising in nonlinear and complex systems, including optics, fluid dynamics, Bose-Einstein condensates and finance and so on.

1 Introduction

Breathers and rational solutions motivate rogue-wave studies, while higher-order vector nonlinear Schrödinger rogue waves remain a target. This paper extends the Manakov-system results to general Nth-order solutions and explores their interactions with solitons.

1 Introduction

  • Breathers have been invoked as rogue-wave models because they are localized in time or space while periodic in the other variable.The introduction notes their study in water waves and optical fibres.Source: §1 · pp. 1–2
  • Higher-order rational solutions are of interest as rogue-wave models because they may have higher amplitudes than the Peregrine breather.Earlier work obtained such solutions using Darboux, Hirota bilinear, generalized Darboux, and algebraic-geometric methods.Source: §1 · p. 2
  • The paper studies the self-focusing VNLSE, whose wave-envelope components have applications including nonlinear optics and Bose-Einstein condensates.The system is also called the Manakov system.Source: §1 · p. 2
  • The study aims to construct general Nth-order rogue-wave solutions of the vector nonlinear Schrödinger equation, extending earlier fundamental and low-order results.The authors report that first-, second-, and third-order VNLSE rogue waves had already been explicitly presented.Source: §1 · pp. 2–3
  • The paper’s solutions combine rational and exponential functions, yielding coexistence patterns of higher-order rogue waves, multi-rogue waves, and bright-dark multisolitons.The introduction identifies rogue-wave and soliton interactions as a source of complex dynamics.Source: §1 · p. 3

2 Asymptotic expansion of Darboux-dressing transformation

The section develops an asymptotic-expansion result that lets the Darboux-dressing construction generate solutions repeatedly at the same spectral parameter. It establishes the transformed eigenfunctions and fields as solutions of the Lax pair and VNLSE.

2 Asymptotic expansion of Darboux-dressing transformation

  • The construction uses the VNLSE’s integrability, expressed through compatibility of its Lax pair, and a Darboux transformation that updates the field matrix.The transformation is given in terms of a fundamental Lax-pair solution and a complex spectral parameter.Source: §2 · pp. 3–4
  • A zero seed typically generates multisolitons and a plane-wave seed breathers, providing distinct starting points for Darboux hierarchies.The section notes that the same-spectral-parameter expansion is introduced to overcome the iteration limitation.Source: §2 · p. 4
  • The expansion theorem addresses the inability to iterate the Darboux-dressing transformation continuously at one spectral parameter by generating new solutions there.It expands a Lax-pair solution near λ1 in powers of δ and constructs iterates from the expansion coefficients.Source: §2 · pp. 4–5
  • The theorem’s induction establishes that the iterated vectors remain solutions of the Lax pair and that the associated transformed fields satisfy the VNLSE.The proof begins with the zeroth-order solution and completes the induction for the Darboux-dressing step.Source: §2 · pp. 3, 6, 9

3 Exact breather solutions of the Lax pair system

Using the plane-wave solution as a seed in the Darboux transformation, the authors construct exact VNLSE breather solutions. Their periodicity depends on the spectral parameter, and a bright-dark rogue wave can emerge during soliton propagation.

3 Exact breather solutions of the Lax pair system

  • The plane-wave seed and Darboux transformation yield exact VNLSE breather solutions built from hyperbolic and exponential functions.For λ = ih, solutions are spatially periodic when −2ω < h < 2ω and temporally periodic when h > 2ω or h < −2ω.Source: §3 · pp. 9, 12
  • The construction includes temporally periodic breathers and spatially periodic breathers, with both bright-dark and typical examples shown.Figures 1 and 2 show temporal periodicity; Figures 3 and 4 show spatial periodicity.Source: Figure 1 · p. 11, Figure 2 · p. 11, §3 · p. 12
  • Requiring the seed-vector components to satisfy a constraint eliminates the exponential functions, yielding dynamics similar to NLSE Akhmediev and Kuznetsov-Ma breathers.The Peregrine breather is described as the limiting case when either periodic solution's period becomes infinite.Source: §3 · p. 12
  • A bright-dark rogue wave emerges during propagation of a bright-dark soliton in the illustrated solution.The text notes that nonzero wavenumber slightly changes the soliton's velocities and phases compared with zero wavenumber.Source: Figure 5 · p. 13, §3 · p. 14

4 Derivation of Nth-order rogue waves solutions

The expansion method constructs Nth-order VNLSE rogue waves in a separated form, with parameter constraints selecting a pure-rational subset. That subset recovers the N=1, 2, and 3 cases previously considered by Zhai et al..

4 Derivation of Nth-order rogue waves solutions

  • The derivation expands around δ = 0 after setting the zero-wavenumber case k = 0 and λ = iω(1 + δ), with 0 < δ < 1.Taylor expansions of the trigonometric and exponential functions are used to expand matrix F.Source: §4 · p. 14
  • The construction takes Z0 as a polynomial in δ whose coefficient vectors Wk are arbitrary constants.The supplied passages also give notation for the expansion coefficients and state that n and m are nonnegative integers with n ≥ m.Source: §4 · pp. 14–15
  • The Nth-order solutions have a separation-of-variables form because the F matrices are independent of t and the G matrices are independent of x.They are obtained by taking λ1 = iω in the expansion theorem.Source: §4 · p. 15
  • Imposing the stated conditions on the Wk components reduces the Nth-order solution to a pure-rational solution.For N = 1, 2, and 3, these solutions match the cases considered by Zhai et al..Source: §4 · p. 15

5 Dynamics of rogue wave solutions

The dynamics vary with the free parameters: second-order rogue waves can coexist with solitons, and higher orders extend these patterns. Parameter choices also shape component amplitudes, breathing behavior, and the timing and location of intensity humps.

5 Dynamics of rogue wave solutions

  • When a1µ30 − a2µ20 = 0 and a1µ31 − a2µ21 = 0, the first-order expression gives a trivial generalization of the rational Peregrine solution, with u1 proportional to u2.The authors focus on other parameter values.Source: §5 · p. 17
  • With W1 = 0, two solitons coexist with a second-order rogue wave whose |u1| peak is about 3 and whose |u2| peak reaches 4 times the background plane-wave amplitude.With W1 = −W0, the rogue wave in u1 is difficult to see, while u2 approaches 5 times the background amplitude; peak amplitudes depend on the free parameters.Source: §5 · p. 17
  • Second-order solutions also show breathing bright and dark solitons when a1 ≠ 0, and one illustrated pattern has five intensity humps at different times or locations.The observed patterns vary across the parameter choices shown in Figures 6 to 10.Source: §5 · p. 17
  • For higher orders, all nonzero ai and Wi produce breathing soliton-like waves, while nonproportional Wi and Wj produce intensity humps at different times or locations.The authors state that the results extend to fourth- and higher-order rogue waves, and that multisoliton velocities generally depend on time.Source: §5 · p. 17

6 Conclusion

The paper constructs higher-order rogue waves in the Manakov system with a Darboux-dressing transformation and asymptotic expansion. The solutions have 3N parameters and exhibit varied dynamics, including rogue waves interacting with multisolitons.

6 Conclusion

  • The Nth-order solutions have 3N parameters, an explicit separation-of-variables form, and patterns including bright-dark higher-order and multi-rogue waves interacting with multisolitons.The paper presents these spatial-temporal patterns as evidence of rich dynamics in the vector system.Source: §6 · pp. 17, 22
  • Figures 6 and 7 depict second-order rogue waves propagating with two bright-dark solitons under different parameter choices.Figure 6 sets µ30 = 1 and µ11 = µ21 = µ31 = 0; Figure 7 sets µ30 = 100 and µ11 = µ21 = −1, µ31 = −100.Source: Figure 6 · p. 18, Figure 7 · p. 18
  • Figures 10–12 show multi-rogue-wave or third-order rogue-wave patterns in propagation with two or three bright-dark solitons.Figure 10 shows multi-rogue-wave patterns with two solitons; Figures 11 and 12 show third-order rogue waves with three.Source: Figure 10 · p. 20, Figure 11 · p. 20, Figure 12 · p. 21
  • The displayed solutions use wavenumber k = 0, although the same procedure can be used for k ≠ 0.The paper also identifies rogue waves in the normal-dispersion regime and Matrix nonlinear Schrödinger equation as topics for future work.Source: §6 · p. 22
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