In this paper, we undertake a systematic exploration of soliton turbulent phenomena and the emergence of extreme rogue waves within the framework of the one-dimensional fractional nonlinear Schrödinger (FNLS) equation, which appears in many fields, such as nonlinear optics, Bose–Einstein condensates, plasma physics, etc. By initiating simulations with a plane wave modulated by small noise, we scrutinized the universal regimes of non-stationary turbulence through various statistical indices. Our analysis elucidates a marked increase in the probability of rogue wave occurrences as the system evolves within a certain range of Lévy index , which can be ascribed to the broadened modulation instability bandwidth. This heightened probability of extreme rogue waves is corroborated through multiple facets, including wave-action spectrum, fourth-order moments, and probability density functions. However, it is crucial to acknowledge that a decrease in also results in a reduction in the propagation speed of solitons within the system. Consequently, only high-amplitude solitons with non-zero background are observed, and the occurrence of collisions that could generate higher-amplitude rogue waves is suppressed. This introduces an inverse competitive mechanism: while a lower expands the bandwidth of modulation instability, it concurrently impairs the mobility of solitons. Our findings contribute to a deeper understanding of the mechanisms driving the formation of rogue waves in nonlinear fractional systems, offering valuable insights for future theoretical and experimental studies.
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January 2025
Research Article|
January 13 2025
The fractional nonlinear Schrödinger equation: Soliton turbulence, modulation instability, and extreme rogue waves
Ming Zhong
;
Ming Zhong
(Conceptualization, Formal analysis, Investigation, Methodology, Software, Writing – original draft)
1KLMM,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
, Beijing 100190, China
2
School of Mathematical Sciences, University of Chinese Academy of Sciences
, Beijing 100049, China
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Weifang Weng;
Weifang Weng
(Conceptualization, Formal analysis, Methodology, Writing – review & editing)
3
School of Mathematical Sciences, University of Electronic Science and Technology of China
, Chengdu 611731, China
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Boling Guo;
Boling Guo
(Conceptualization, Formal analysis, Methodology, Writing – review & editing)
4
Institute of Applied Physics and Computational Mathematics
, Beijing 100088, China
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Zhenya Yan
Zhenya Yan
a)
(Conceptualization, Formal analysis, Funding acquisition, Methodology, Supervision, Writing – review & editing)
1KLMM,
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
, Beijing 100190, China
2
School of Mathematical Sciences, University of Chinese Academy of Sciences
, Beijing 100049, China
a)Author to whom correspondence should be addressed: [email protected]
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a)Author to whom correspondence should be addressed: [email protected]
Chaos 35, 013131 (2025)
Article history
Received:
October 02 2024
Accepted:
December 16 2024
Citation
Ming Zhong, Weifang Weng, Boling Guo, Zhenya Yan; The fractional nonlinear Schrödinger equation: Soliton turbulence, modulation instability, and extreme rogue waves. Chaos 1 January 2025; 35 (1): 013131. https://doi.org/10.1063/5.0242142
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