Interaction dynamics of hybrid solitons and breathers for extended generalization of Vakhnenko equation
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ORIGINAL PAPER
Interaction dynamics of hybrid solitons and breathers for extended generalization of Vakhnenko equation Bang-Qing Li · Yu-Lan Ma
Received: 22 August 2020 / Accepted: 12 October 2020 © Springer Nature B.V. 2020
Abstract The main attention of this study is focused on the interaction dynamics of breather, and hybrid solitons and breather for an extended generalization of Vakhnenko equation. The general form of the N -order auxiliary function is first derived by the Hirota bilinear method. Then, the N -order solutions can be obtained. When N ≥ 2, the loop-like breather may emerge by taking the dispersion coefficients as conjugate complex number. The breather is like a magical spring with good elasticity, changeable period and thickness. Furthermore, novel interaction features between breathers, between soliton/solitons and breather/breathers, are observed by visualizing method. The results show that there are abundant dynamics during the interactions, such as elastic collision, phase shift, amplitude amplification, collapse and bulge effect. Keywords Extended generalization of Vakhnenko equation (EGVE) · Hirota bilinear method · General B.-Q. Li Academy of Systems Science, Beijing Technology and Business University, Beijing 100048, People’s Republic of China e-mail: [email protected] B.-Q. Li School of Computer, Beijing Technology and Business University, Beijing 100048, People’s Republic of China Y.-L. Ma (B) School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048, People’s Republic of China e-mail: [email protected]
N -order auxiliary function · Loop-like breather · Loop-like soliton · Interactions dynamics
1 Introduction Nonlinear evolution equations (NLEEs), involved both space and time variables, are fundamental models in nonlinear science, especially in nonlinear dynamics. Among these models, there are a few of NLEEs which possess loop-like wave solutions, such as shortpulse equation [1–3], Vakhnenko equation (VE, a high-frequency wave equation) [4–7] and Kraenkel– Manna–Merle equation (an ultra-short wave equation in ferrites) [8–11]. Loop-like wave solutions can be expressed mathematically as multiple-valued function, and also describe a special physical phenomenon where waves may be compressed in spatiotemporal domain. These waves have shown great potential application prospects in ultra-fast optics [12]. With the increasing demand for high-speed communication, some new concepts, such as 5G even 6G mobile networks, ultra-fast optics and attosecond physics, have been emerging for the fact that the tools and techniques are now becoming available for steering and tracing electronic motion in atoms, molecules and nanostructures [13–15]. Limited by the narrower space and shorter time, the solitons may be squeezed and even compressed into higher frequency and/or shorter waves. It is natural that the NLEEs with the looplike soliton solutions become ideal models to depict
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the wave propagations in new communication systems [16–1
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