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N-Fold generalized Darboux transformation and semirational solutions for the Gerdjikov-Ivanov equation for the Alfvén waves in a plasma

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Abstract

Alfvén waves propagating parallel to the ambient magnetic field are modeled via the Gerdjikov-Ivanov equation. With respect to the transverse magnetic field perturbation for such an equation, we derive an N-fold generalized Darboux transformation and some semirational solutions on the constant/periodic background, where N is a positive integer. Such semirational solutions consist of the l-th-order rogue waves, the \(\left( \kappa -l-r\right) \)-th-order nondegenerate breathers and the r-th-order degenerate breathers, where \(\kappa =2,3,\ldots ,N\), \(l=0,1,2,\ldots ,\kappa -1\), and \(r=0,2,3,\ldots ,\kappa \). With some conditions, the l-th-order rogue wave is close to the \(\frac{1}{2}(\kappa -l)(\kappa +l+1)\) elements of the \(\left( \kappa -l\right) \)-th-order breathers which locate in the center of the \(x\) \(-\) \(t\) plane, and then the interaction among them forms the \(\kappa \)-th-order rogue wave, where x and t denote the space and time coordinates, respectively. Through the asymptotic analysis, we find that the trajectories of the second-order degenerate breathers are the logarithmic curves, and the asymptotic breathers are in good coherence with the corresponding analytic solutions in the far-field region of the x–t plane. Interaction between the second-order nondegenerate breathers and the first-order rogue wave reveals the periodic attraction and repulsion.

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Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Notes

  1. Solutions corresponding to the same spectral parameter have been called degenerate solutions, while the solutions corresponding to the different spectral parameters have been called nondegenerate solutions [26, 27].

  2. Different from the N-fold GDT in Ref. [37] which can iterate k times under the same spectral parameter, N-fold GDT for Eq. (1), to be done in this paper, can iterate \(m_k\) times under each of the k different spectral parameters, where \(k=1,2,\ldots ,n\), n and \(m_k\)’s are the positive integers, and \(\sum \limits _{k=1}^{n}m_k=N\).

  3. We can analyze characteristics of the three types of the third-order semirational solutions on the periodic background in the similar way.

Abbreviations

\({\mathbb {C}}\), \({\mathbb {R}}\) and \({\mathbb {I}}\) :

Sets of complex, real and pure imaginary numbers, respectively

Re and Im :

Corresponding real and imaginary parts of a complex scalar, respectively

\(\delta _k^b\) :

Kronecker delta function, i.e., \(\delta _k^b={\left\{ \begin{array}{ll} 0,&{}k\ne b\\ 1, &{}k=b \end{array}\right. }\)

\(*\) :

Complex conjugate

T:

Vector transpose

\(\det \) :

Determinant of a matrix

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Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023, 11805020 and 11471050, by the Fund of State Key Laboratory for Information Photonics and Optical Communications (Beijing University for Posts and Telecommunications) (IPOC:2017ZZ05), by the Fundamental Research Funds of the Central Universities of China under Grant No. 2011BUPTYB02, and by the Beijing University of Posts and Telecommunications Excellent Ph.D. Students Foundation (No. CX2021318).

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  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Su-Su Chen, Bo Tian, He-Yuan Tian & Dan-Yu Yang

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Chen, SS., Tian, B., Tian, HY. et al. N-Fold generalized Darboux transformation and semirational solutions for the Gerdjikov-Ivanov equation for the Alfvén waves in a plasma. Nonlinear Dyn 108, 1561–1572 (2022). https://doi.org/10.1007/s11071-021-07183-8

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