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

Lou, Yu; Zhang, Yi; Ye, Rusuo; Li, Miao

doi:10.1016/j.wavemoti.2021.102795

Cited by 15 publications

(3 citation statements)

References 33 publications

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“…In general, the DT cannot be directly used to construct rational solutions for evolution equations. In [16], Gue et al proposed a simple method (the modified DT) which can be applied to the GI equation for constructing the higher-order rogue wave solutions [17,32].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Binary Darboux transformation for the Gerdjikov–Ivanov equation

Yilmaz

2022

Wave Motion

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Section: Conclusion and Discussionmentioning

confidence: 99%

Binary Darboux transformation for the Gerdjikov–Ivanov equation

Yilmaz

2022

Wave Motion

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“…A new direct approach namely exp(− ( ))-expansion method was employed to seek exact solutions of GIE which include variety of models (Kadkhoda and Jafari 2017). The modulation instability and higherorder rogue waves for the GIE were investigated in Lou et al (2021). The multi-component nonlocal reverse-time GIE was derived through nonlocal group reduction of the multicomponent GIE (Zhang and Dong 2022).…”

Section: Introductionmentioning

confidence: 99%

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Abdel‐Gawad

2023

Opt Quant Electron

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The Gerdjikov–Ivanov equation (GIE) occupied a remarkable area of research in the literature. In the present work, a modified GIE (MGIE) is considered which is new and was not studied in the literature. Also, the modified-unified method (MUM) is used to obtain approximate analytic solutions (AASs) of MGIE. Up to our knowledge, no AASs for non-integrable complex field equation were found up to now. Thus the AASs found, here, are novel. The UM addresses finding the exact solutions to integrable equations. In this sense as no exact solution for MGIE exists, consequently, it is not integrable. So, here, approximate analytic optical soliton solutions are invoked. The UM stands for expressing the solution of nonlinear evolution equations in polynomial and rational forms in an auxiliary function (AF) with an appropriate auxiliary equation. For finding exact solutions by the UM, the coefficients of the AF, with all powers, are set equal to zero, For a non-integrable equation, only approximate solutions are affordable. In this case, we are led to utilizing the MUM. Herein, non-zero coefficients (residue terms (RTs)) are considered as errors, which are space and time-independent. It is worth mentioning that, this is in contrast to the errors found by the different numerical methods, where they are space and time-dependent. Further, in the present case, the maximum error is controlled via an adequate choice of the parameters in the RTs. These solutions are displayed in graphs. Breather soliton, chirped soliton and M-shape soliton, among others, are observed. Furthermore, modulation instability (MI) is studied and it is found MI triggers when the coefficient of the nonlinear dispersion exceeds a critical value.

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“…这里 , 其中表示空间坐标, 表示时间坐标, 符号*表示复共轭, 下标表示偏导数. 已经有许多学者对此方程进行了研究, 并推导出在零背景下的呼吸子 [21][22][23] . 对于此方程, Fan [24,25] 给出了双哈密顿结构、Liouvill可积性和代数几何解.…”

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Breathers of Gerdjikov-Ivanov equation under background of elliptic function

Yao,

Zhang,

Xiong

2024

Acta Phys. Sin.

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As one specific type of local solutions of nonlinear evolution equation, the breathers have the characteristic of envelope oscillation structure. This kind of oscillation is periodic. According to the periodicity of the distribution and evolution directions, there are three kinds of breathers, namely, the Kuznetsov-Ma breather (KMB), the Akhmediev breather (AB) and the general breather (GB). In recent years, the propagation of envelope breathers on the periodic background has been observed in many nonlinear physical fields, including nonlinear optical fibers and hydrodynamics. It is believed that the breathers can arise due to the modulational instability of the periodic waves, and they demonstrate many rich physical properties and dynamic behaviors of interactions. Therefore, recently great attention has been paid to the breathers on the periodic background in nonlinear science. As an important integrable model, the Gerdjikov-Ivanov (GI) equation can be used to describe various nonlinear phenomena in many physical fields such as in the quantum field theory, weak nonlinear dispersive water wave and nonlinear optics. It is very meaningful to solve various types of solutions of this model to describe the propagation of nonlinear waves. As far as we know, the breather solutions for the GI equation have not been constructed on the elliptic function background. In this paper, firstly, elliptic function solutions of the GI equation are solved by the modified squared wave (MSW) function approach and the traveling wave transformation. Then, we obtain the basic solution of the Lax pair corresponding to the Jacobi elliptic function seed solution. Based on the elliptic function transformation formulas and the integral formulas, the potential function solution can be expressed in terms of the Weierstrass elliptic function. Secondly, by the once iterated Darboux transformation, three types of breather solutions on the elliptic function background are constructed including the GB, the KMB and the AB. In addition, we analyze the dynamic behaviors of these three kinds of breathers, and present their three-dimensional space-time structures. By the twice iterated Darboux transformation, on the dn periodic background we exhibit three types of interactions between two breathers, i.e., a GB and a KMB, an AB and a KMB, a GB and an AB. Finally, we also present three types of interactions between two breathers on the general periodic background.

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

Cited by 15 publications

References 33 publications

Binary Darboux transformation for the Gerdjikov–Ivanov equation

Binary Darboux transformation for the Gerdjikov–Ivanov equation

Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Breathers of Gerdjikov-Ivanov equation under background of elliptic function

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