Modulation instability, higher-order rogue waves and dynamics of the Gerdjikov–Ivanov equation

doi:10.1016/j.wavemoti.2021.102795

Wave Motion

Volume 106, November 2021, 102795

https://doi.org/10.1016/j.wavemoti.2021.102795 Get rights and content

Abstract

We investigate the modulation instability and higher-order rogue waves for the Gerdjikov–Ivanov equation. Based on the theory of the linear stability analysis, the modulation instability is the condition of the existence of the rogue waves. With the help of Darboux transformation and a variable separation technique, the formula of the higher-order rogue wave solutions is given explicitly. The kinetics of the first-, second-, and third-order rogue wave solutions are elucidated from the viewpoint of three-dimensional structures. More specifically, it is shown that this method is fairly powerful and handy to obtain the higher-order rogue wave solutions which appear in different phenomena in applied sciences and mathematical physics.

Introduction

A special solitary wave, rogue wave, has been extensively investigated in many fields, such as plasmas, oceanic waves and optics(see [1], [2], [3]). Rogue waves are characterized by space–time localness [4] and usually are described by rational solutions of the nonlinear Schrödinger equation [5], [6], [7]. The generation of the rogue wave is closely related to modulation instability (MI). The MI plays a prominent position in diverse fields, for example, plasma physics, nonlinear optics and so on [8], [9], [10], [11].

As an integrable generalization of the nonlinear Schrödinger equation, derivative nonlinear Schrödinger (DNLS) equations have three generic deformations, the DNLS-I equation [12], [13] $i q_{t} - q_{x x} + i {(q^{2} q^{*})}_{x} = 0,$ the DNLS-II equation [14], [15] $i q_{t} + q_{x x} + i q q^{*} q_{x} = 0,$ and the DNLS-III equation or Gerdjikov–Ivanov (GI) equation [16] $i q_{t} + q_{x x} - i q^{2} q_{x}^{*} + \frac{1}{2} q^{3} {(q^{*})}^{2} = 0 .$ The DNLS equations have a great deal of applications in nonlinear optics fibers [17], [18], [19] and plasmas [20]. Moreover, DNLS equations can be transformed into each other by a chain of gauge transformations and the method of the gauge transformation can also be applied to some generalized cases [21], [22], [23].

In recent years, various methods have been proposed for getting exact solutions of GI equation, such as the algebro-geometric method [24], Riemann–Hilbert method [25] and Darboux transformation (DT) method [26], [27], [28], [29]. Notably, the DT method has been proved to be a powerful method for obtaining soliton, breather and rogue wave solutions [30], [31], [32], [33]. At present, the fundamental rogue wave has been generalized to higher order in [34], [35], [36], [37]. However, there is not a straightforward extension to construct the higher-order rogue wave solutions at the same eigenvalue. It is extremely tough and challenging to obtain the higher-order rogue wave solutions via generalized DT [37] when the order of the rogue wave solutions increases. Therefore, one will utilize a distinct technique (see [38], [39]) from generalized DT which can precisely avoid these shortcomings. Using this method, the family of the solution of the Lax pair is written through the exponential matrix in a variable separation form. What is more important, it not only simplifies the calculation process, but also makes the selection of parameters easier.

The rest of this paper is as follows. In Section 2, we introduce the Lax pair and the DT method of the GI equation. In Section 3, we investigate the linear stability of a plane wave solution regarding to MI. In Section 4, the higher-order rogue wave solutions in a general formula are presented and their dynamic behaviors are analyzed. The discussion and conclusion are located in the final section.

Section snippets

Darboux transformation

Let us begin with the coupled GI equations, $i q_{t} + q_{x x} + i q^{2} r_{x} + \frac{1}{2} q^{3} r^{2} = 0,$ $i r_{t} - r_{x x} + i r^{2} q_{x} - \frac{1}{2} q^{2} r^{3} = 0,$ which are readily reduced to Eq. (3) for $r = - q^{*}$ . The Lax pairs of coupled GI Eqs. (4) are given by the GI system with a quadratic spectral [16], $Ψ_{x} = U Ψ, U = - i σ λ^{2} + U_{1} λ + U_{0},$ $Ψ_{t} = V Ψ, V = - 2 i σ λ^{4} + V_{3} λ^{3} + V_{2} λ^{2} + V_{1} λ + V_{0},$ where $σ = (\begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix}), U_{1} = (\begin{pmatrix} 0 & q \\ r & 0 \end{pmatrix}), U_{0} = (\begin{pmatrix} - \frac{i}{2} q r & 0 \\ 0 & \frac{i}{2} q r \end{pmatrix}),$ $V_{3} = 2 U_{1}, V_{2} = - i σ q r, V_{1} = (\begin{pmatrix} 0 & i q_{x} \\ - i r_{x} & 0 \end{pmatrix}),$ $V_{0} = (\begin{pmatrix} \frac{1}{2} (r q_{x} - q r_{x}) + \frac{1}{4} i q^{2} r^{2} & 0 \\ 0 & - \frac{1}{2} (r q_{x} - q r_{x}) - \frac{1}{4} i q^{2} r^{2} \end{pmatrix}) .$

Here, the spectral parameter $λ$ is an arbitrary complex constant. The compatibility condition $U$

Modulation instability

MI is the basic mechanism for forming rogue wave solutions. Therefore, before studying higher-order rogue wave solutions, we consider MI of Eq. (3).

It is readily shown that the GI equation (3) admits a seed solution $q [0] = c e^{i θ}, θ = a x + (\frac{c^{4}}{2} - c^{2} a - a^{2}) t,$ where $a$ and $c$ are arbitrary real constants. Then the small perturbation is added into the plane wave $q = (1 + \tilde{q}) q [0],$ where $\tilde{q}$ is a function of $x$ and $t$ . The substitution of Eq. (11) into Eq. (3) yields the linearized equation $i {\tilde{q}}_{t} + {\tilde{q}}_{x x} + 2 i a {\tilde{q}}_{x} - i c^{2} {\tilde{q}}_{x}^{*} + (c^{4} - c^{2} a)$

Higher-order rogue wave solutions

Resorting to the DT method, one will arrive at the explicit formula of different types of the higher-order rogue wave solutions. According to Ref. [38], the corresponding solution of (5) can be sought in the following form $Ψ = (\begin{matrix} φ (x, t) \\ ϕ (x, t) \end{matrix}) = AFGZ,$ where $A = (\begin{pmatrix} 1 & 0 \\ 0 & e^{- i θ} \end{pmatrix}),$ $F = exp (i Θ x), G = exp (i Υ t),$ with $θ = i a x + i (\frac{c^{4}}{2} - c^{2} a - a^{2}) t .$

Meanwhile, it is required that the matrices $Θ$ , $Υ$ meet the commutator relationship $[Θ, Υ] = Θ Υ - Υ Θ = 0 .$

Next, substituting (17) into (5) generates $A_{x} + i A Θ - U A = 0, A_{t} + i A Υ - V A = 0 .$

After that, it can be

Conclusion

In this paper, resorting to the linear analysis theory, we study MI which is the condition of the existence of the rogue wave. Then we explicitly express the higher-order rogue wave solutions of GI equation without a lot of tedious calculations by the DT method and the variable separation technique. It can be observed that changing the parameters results in the higher-order rogue waves exhibiting distinct phenomena from [28], [29] which promote our knowledge of rogue wave phenomena.

CRediT authorship contribution statement

Yu Lou: Conceptualization, Methodology, Visualization, Writing - original draft. Yi Zhang: Supervision, Writing - review & editing. Rusuo Ye: Software, Validation. Miao Li: Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11371326 and No. 11975145).

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Citation Excerpt :
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Modulation instability, higher-order rogue waves and dynamics of the Gerdjikov–Ivanov equation

Abstract

Introduction

Section snippets

Darboux transformation

Modulation instability

Higher-order rogue wave solutions

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Physica D

Chaos Solitons Fractals

Nonlinear Anal. RWA

Wave Motion

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Appl. Math. Comput.

Surface plasma rogue waves

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Oceanic rogue waves

Annu. Rev. Fluid Mech.

Optical rogue waves

Nature

Rogue waves and rational solutions of the nonlinear Schrödinger equation

Phys. Rev. E

Water waves, nonlinear Schrödinger equations and their solutions

J. Aust. Math. Soc. Ser. B

Rogue waves and rational solutions of the nonlinear Schrödinger equation

Phys. Rev. E

Rational multisoliton solutions of the nonlinear Schrödinger equation

Sov. Phys. Dokl.

Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma

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Rogue wave modes for a derivative nonlinear Schrödinger model

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