# Editorial: Analytical methods for nonlinear oscillators and solitary waves

^{1}School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an, China^{2}School of Mathematics, China University of Mining and Technology, Xuzhou, China^{3}College of Textile and Clothing Engineering, Soochow University, Suzhou, China^{4}Mechanical Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran^{5}Mechanical Engineering Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt^{6}Department of Structural Analysis, Technical University of Berlin, Berlin, Germany^{7}Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia

**Editorial on the Research Topic ****Analytical methods for nonlinear oscillators and solitary waves**

## 1 Introduction

Physics is mathematical, new physical phenomena require new mathematical tools. This Research Topic is an attractive introduction of some new mathematical concepts, e.g., the two-scale fractal geometry, the fractal-fractional models, the homotopy perturbation method and the frequency formulation appliable to physics. The new findings of physical phenomena using the new mathematical tools have excited physicists with their potential to reveal secrets in physics and have triggered new research frontiers in physics. Here is an overview of the Research Topic.

## 2 Fractional soliton vs. fractal soliton

Fractional soliton is a new concept in mathematics, in this paper, Zeng et al. studied the fractional Kdv–Burgers equation to reveal that the soliton profile is not function of *t* and *x*, but *t*^{η} and *x*^{η}, where η is the fractional order.

Fractal soliton, on the other hand, is a solitary wave moving along an unsmooth boundary or through a porous medium [1]. When an attosecond electron beam is trapped in and propagate with the laser pulse, the travelling solitary wave can be modelled in a fractal space [2], and the attosecond physics won 2023 Nobel Prize in Physics [3]. Discontinuous time appears on an attosecond (10^{−18} s) scale, so fractal time has to be adopted [4, 5], and pinpointing the fractal dimensions is tricky, especially when the studied system has not seeming self-similarity, now He-Liu’s fractal dimensions formulation [6] makes the fractal theory accessible to porous media and discontinuous time.

## 3 Fractional vibration vs. fractal vibration

The fractional calculus can also be applied to model the memory property of a damped vibration system. In this paper, Zhang et al. studied a fractional stochastic vibration system by taking full advantage of the memory property of the Caputo fractional derivative.

A fractal vibration system, on the other hand, works in a fractal space. The fractal vibration theory allows scientists to insight into the vibration properties on a molecule scale. The traditional vibration theory cannot model the effects of molecules or nanoparticles’ size and distribution in air on the vibrating properties. Tian et al. [7] considered the effect of the air pollution on the operation of the MEMS system, and concluded that the fractal dimensions can be used for controlling the pull-in instability. In this paper, Lin and Li applied the fractal vibration theory to elucidate the ions release mechanism instead of the traditional diffusion process, opening up a flood of promising opportunities to design new hollow fibers.

## 4 Homotopy perturbation method

The homotopy perturbation method (HPM) was proposed by Ji-Huan He [8], a heuristic review on the method is available in Ref. [9]. In an interview with ScienceWatch.com on February 2008, Ganji http://archive.sciencewatch.com/dr/fbp/2008/08febfbp/08febGanji/ emphasized the homotopy perturbation method (HPM), “wherever a nonlinear equation is found, Dr. He’s HPM will be the primary tool of discovery,” and he further concluded, “He’s homotopy perturbation method itself is mathematically beautiful and extremely accessible to non-mathematicians.” In the last two decades, Dr. Ganji’s prediction is coming true. There are many modifications of the homotopy perturbation method, among which He-Laplace method is extremely suitable for fractional calculus [10–12], and Li-He’s modified homotopy perturbation method [13–15] for forced oscillators.

In this paper, Qayyum et al. found that the homotopy perturbation method is extremely suitable for the search for fractional soliton solutions, Tao et al. coupled the Aboodh transformation with the homotopy perturbation method, a new hope for fractional calculus, Buhe et al. applied the method to study forest resource and there is the possibility to extend it to other natural resources, especially the grassland resources.

## 5 Frequency formulation

The simpler is the better for most physical problems. So far the simplest approach to a nonlinear oscillator is He’s frequency formulation [16–18]. There are many modifications, the most famous one is the Hamiltonian-based frequency-amplitude formulation [19, 20]. El-Dib extended it to time-delayed vibration systems [21]. In this paper, Niu et al. extended the frequency formulation to fractal–fractional non-linear oscillators.

## 6 Concluding remarks

This Research Topic of *Frontiers in Physics* consists mainly of a Research Topic of mathematics methods appliable to physics, it is to bring to the fore the many new and exciting applications of some new mathematical theories of the two-scale fractal theory and the fractal-fractional calculus, it can attract much attention from different fields, such as mathematics, physics, artificial intelligence, neural network, computer science, textile engineering, material science and others. We hope that this Research Topic will prove to be a timely and valuable reference for researchers in this Research Topic.

## Author contributions

C-HH: Writing–original draft, Writing–review and editing. J-HH: Writing–review and editing. HS: Writing–review and editing. YE-D: Writing–review and editing. DM: Writing–review and editing. AA: Writing–review and editing.

## Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

## Publisher’s note

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## References

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2. Duchateau G. Theoretical derivation of laser-dressed atomic states by using a fractal space. *Eur Phys J Plus* (2018) 133(5):186. doi:10.1140/epjp/i2018-12017-y

3. Castelvecchi D, Sanderson K. Physicists who built ultrafast 'attosecond' lasers win Nobel Prize. *Nature* (2023) 622:225–7. doi:10.1038/d41586-023-03047-w

5. He JH. Fractal calculus and its geometrical explanation. *Results Phys* (2018) 10:272–6. doi:10.1016/j.rinp.2018.06.011

6. He CH, Liu C. Fractal dimensions of a porous concrete and its effect on the concrete’s strength. *Facta Universitatis Ser Mech Eng* (2023) 21:137–50. doi:10.22190/fume221215005h

7. Tian D, Ain QT, Anjum N, He CH, Cheng B. Fractal N/MEMS: from pull-in instability to pull-in stability. *Fractals* (2021) 29:2150030. doi:10.1142/s0218348x21500304

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19. Ma HJ. Simplified Hamiltonian-based Frequency-amplitude formulation for nonlinear vibration systems. *Facta Universitatis Ser Mech Eng* (2022) 20(2):445–55. doi:10.22190/fume220420023m

20. He JH, Hou WF, Qie N, Gepreel KA, Shirazi AH, Sedighi HM. Hamiltonian-based Frequency-amplitude formulation for nonlinear oscillators. *Facta Universitatis Ser Mech Eng* (2021) 19(2):199–208. doi:10.22190/fume201205002h

Keywords: soliton, nonlinear osciilation, fractal theory, fractional calculus, homotope perturbations method

Citation: He C-H, He J-H, Sedighi HM, El-Dib YO, Marinkovic D and Alsolami AA (2023) Editorial: Analytical methods for nonlinear oscillators and solitary waves. *Front. Phys.* 11:1309182. doi: 10.3389/fphy.2023.1309182

Received: 07 October 2023; Accepted: 23 October 2023;

Published: 03 November 2023.

Edited and reviewed by:

Alex Hansen, Norwegian University of Science and Technology, NorwayCopyright © 2023 He, He, Sedighi, El-Dib, Marinkovic and Alsolami. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Chun-Hui He, mathew_he@yahoo.com