The Study on the Fundamental of Dark and Bright Solitons

Abstract

The words bright and dark are taken from the field of optics, where they are used to describe the bright spots and black shadows that appear in optical fibres. However, in the 1830s, soliton observations were made in water. Oceanographers were initially taken aback by the discovery in the 1960s and 1970s that bright solitons were present on the surface of deep ocean waters. However, numerous experiments have since been conducted to investigate and validate the phenomenon, some of which have identified bright solitons as the cause of rogue waves at sea. Both bright and dark solitons have now been observed in Bose-Einstein condensates, plasmas, fibre optics, and other environments. [30]

Introduction

I. INTRODUCTION

It is challenging to define soliton in a way that is widely accepted. Three features are offered to solitons by Drazin & Johnson (1989, p. 15). [1]. They have three characteristics: they have a permanent form, they are limited within a region, and they can interact with other solitons. They also avoid phase shifts but otherwise survive collisions unharmed.There are more formal formulations available, but they require a large amount of mathematics. Furthermore, some scientists designate events that do not satisfy these three conditions as soliton phenomena (for instance, the "light bullets" in nonlinear optics are sometimes referred to as solitons, despite the fact that they lose energy during interaction).[2]Wave patterns can be localised and permanent depending on how dispersion and nonlinearity combine. Envision a bright pulse passing through glass. It is possible that this pulse is made up of several different light frequencies. These various frequencies pass through glass at different speeds due to its dispersion, which changes the pulse's structure over time. There is also the nonlinear Kerr effect, in which the brightness or amplitude of the light determines a material's refractive index at a certain frequency. Because of the precise cancellation of the dispersion effect by the Kerr effect, a properly generated pulse maintains its shape over time. The pulse is a soliton as a result [3]. Solitons solve a wide range of entirely solvable models, including as the coupled nonlinear Schrödinger equation, the sine-Gordon equation, the Korteweg–de Vries equation, and several nonlinear Schrödinger equations. The soliton solutions are stable because of the integrability of the field equations, which is usually achieved using the inverse scattering transform. One active and expansive area of mathematics is the mathematical theory of these equations.[3] In some "undular" tidal bore occurrences occurring in a few rivers, like the River Severn, a train of solitons moves in tandem with a wavefront. As internal waves propagate down the oceanic pycnocline, propelled by the seafloor's topography, more solitons are created. Additional atmospheric solitons exist. Consider the morning glory cloud in the Gulf of Carpentaria as an example. It is created by massive linear roll clouds caused by pressure solitons travelling across a temperature inversion layer. Pressure solitons are employed in the recently developed, but not generally recognised, soliton model of neuroscience to explain the signal conduction within neurons [3].Any solution of a system of partial differential equations that is persistent against decay to the "trivial solution" is referred to as a topological soliton, also known as a topological defect. The origin of soliton stability is not the integrability of the field equations, but topological constraints. Since the differential equations have to satisfy a set of boundary conditions and preserve the nontrivial homotopy group of the boundary, constraints are nearly often present. Therefore, solutions to differential equations can be grouped using homotopy classes [3].

To be more precise, soliton transmission in optical fibres enhances data transmission quality by mitigating two main forms of pulse degradation. The dispersion that occurs when pulses travel over lengthy fibre sections is one kind of deterioration. The other is the nonlinear effects resulting from signals interacting with the fibre and with each other in a power-dependent manner. In general, the two effects compound each other to worsen the condition; however, for some forms and powers of optical pulses, the effects can cancel each other out, at least to first-order approximation. These kinds of pulses are referred to as soliton pulses.One of the key benefits of solitons for long-distance, high-speed gearboxes is their inherent longevity. Despite soliton attenuation, soliton can be made intrinsically stable over long fibre lengths.

This provides a means of mitigating the degradation of signal quality caused by dispersion and nonlinear effects, which is a significant issue at 10 Gbit/s and becomes worse at higher transmission speeds. Because of these characteristics, researchers are working on soliton systems for planned 40-Gbit/s and long-haul 10-Gbit/s systems [4].

II. LITERATURE REVIEW

After seeing a single wave on Scotland's Union Canal in 1834, John Scott Russell (1808–1882) became the first person to record the soliton phenomenon. He named the phenomenon the "Wave of Translation" when he was able to replicate the event in a wave tank. Similar waves to those Russell saw can be described by locally limited, strongly stable propagating solutions to the Korteweg–de Vries equation. Zabusky and Kruskal were the ones who first gave these solutions the name "soliton." With the suffix "on" reflecting its original usage to represent particles such as hadrons, baryons, and electrons and reflecting their observed particle-like activity, the name was intended to describe the solitary nature of the waves.[3].

A young engineer named John Scott Russell, engaged for a summer project in 1834 to look at ways to enhance the designs for barges intended to traverse canals—specifically, the Union Canal in Edinburgh, Scotland—made the first reported observation of a lone wave. The barge came to a sudden stop one August day when the tow line holding the mules to the vessel broke. Still, the mass of water ahead of the blunt prow of the barge rolled forward rapidly, forming a large isolated elevation, a smooth, rounded mound of water, and continuing down the canal without altering its shape or speed. Russell (1844). Russell rode up to the Wave of Translation and passed it after investigating this coincidental discovery further. The Wave of Translation was rolling at around eight or nine miles per hour, keeping its original measurements of roughly thirty feet by one foot to one and a half. Then, in carefully monitored laboratory trials, he employed a wave tank; he published the results in 1844 (Russell, 1844). He gave the following four instances: In single waves, he noticed hyperbolic secant structure. If the initial quantity of water is large enough, it can create two or more waves that progressively split away and become almost solitary.When separated, waves can pass one another "without change of any kind". In a shallow water channel of height h, a single wave with amplitude A travels at the speed of [g(A+h)]1/2, where g is the gravitational acceleration. Greater amplitude waves travel faster than lower amplitude waves in this nonlinear phenomenon. In 1895, the Dutch physicist Diederick Korteweg and his pupil Gustav de Vries (KdV) (de Vries, 1895) (Scott, 2005) created the nonlinear partial differential equation (PDE) that bears their names to this day. Russell's experiments may be explained by the KdV equation (1), according to Korteweg and de Vries. Equation (1) demonstrates that the amplitude effect, a nonlinear component, and the dispersive term—which permits waves of various wavelengths to travel at different velocities—combine to determine the rate of change of the wave's height over time. Russell's wave was matched by a periodic solution discovered by Korteweg and de Vries, in addition to a solitary-wave solution. A trade-off between dispersion and nonlinearity led to these solutions. Until 1965, when mathematicians, physicists, and engineers researching water waves disregarded both Russell's findings and the work of Norman Zabusky and Martin Kruskal, who published their numerical solutions to the KdV equation (and coined the term "soliton") (Zabusky, 1965). On the cubic, Fermi–Pasta–Ulam (FPU) nonlinear lattice, Kruskal obtained a continuous description of the oscillations of unidirectional waves propagating, as stated in (1) (Fermi, 1955; Porter, 2009b; Weissert, 1997). Simultaneously, Morikazu Toda made history by being the first to identify a soliton in the discrete, integrable system that is now known as the Toda lattice (Toda, 1967).[5]

Films of interacting solitary waves in an FPU lattice, the KdV equation, and a modified KdV equation were produced by Gary Deem, Zabusky, and Kruskal in 1965; see the discussion in the review work (Zabusky, 1984). We illustrate the dynamics of solitons in the space-time diagram shown in Figure 1 using the KdV equation. Robert Miura discovered a precise transition between this modified KdV equation and equation (1) when he understood the significance of this discovery (Miura, 1976). Interest in the mathematical study of solitons arose after Clifford Gardner, John Greene, Martin Kruskal, and Robert Miura used the inverse scattering approach to solve the KdV equation's initial-value problem in 1967 (Miura, 1968; Gardner, 1967; Gardner, 1974). As a result, a suitable definition of integrability for continuum frameworks was created. By demonstrating the integrability of the nonlinear Schrödinger (NLS) problem and the availability of soliton solutions, Alexei Borisovich Shabat and Vladimir Zakharov expanded the inverse scattering technique in 1972. One of the various nonlinear PDEs for which Mark Ablowitz, David Kaup, Alan Newell, and Harvey Segur demonstrated that they had soliton solutions in 1973 was the sine-Gordon equation. Thanks to Albert Backlünd's 19th-century studies of surfaces with constant negative Gaussian curvature, this equation was already known to be integrable. Since then, other researchers (in one and several spatial dimensions) have explored related soliton solutions and deduced alternative integrable PDEs.

The Kadomtsev–Petviashvili (KP) equation shows that a more intricate definition of a "soliton" in many spatial dimensions is necessary.

Analytical methods for studying solitary waves in nonintegrable equations usually involve the use of asymptotic analysis, variational approximations, and/or perturbative techniques (Kivshar, 1989). (Scott, 2005) The coupled mode equations of the fibre Bragg grating are a well-known illustration of a nonintegrable system that has exact solutions for solitary waves that are isolated in optics.One of the most active fields in physics and mathematics today is the study of solitons and solitary waves (Scott, 2005). It has affected a wide range of disciplines, including as experimental science and pure mathematics. This has produced significant findings in a number of fields, including integrable systems, biology, optics, supersymmetry, and nonlinear dynamics.

Conclusion

Solitons light pulses propagate without widening over very long distances in a non-linear dispersive medium. They have therefore attracted a lot of interest from the communications industry. Solitons operate in optical fibres because of two effects that, given the right circumstances, can cancel one other out. One of these is chromatic dispersion, which occurs when pulses with various wavelengths disperse as they pass through an optical fibre. Self-phase modulation, or SPM, is the alternative that works over a wider variety of pulse spectrum wavelengths. The pulse can maintain its form or dispersion once it reaches equilibrium in the fibre since dispersion and SPM can cancel one another out. The pulse may potentially more severely expand or compress as a result of SPM. Conversely, attenuation reduces the pulse\'s strength and makes it more difficult for it to maintain its shape over the fibre span. Optical amplifiers have been constructed to balance attenuation and preserve pulse shapes. [24][25]. Solitons are obtained by solving non-linear wave equations that characterise the propagation of waves in certain physical systems. Mathematical models of a variety of systems, including water waves, crystal lattice vibrations, and optical waveguides, show these waves as solutions. On a modulationally stable continuous-wave background, dark solitons exhibit local dips. Dark solitons in fibre lasers are less prone to loss and more stable in noisy environments than dazzling solitons.

References

[1] Drazin, P. G.; Johnson, R. S. (1989). Solitons: an introduction (2nd ed.). Cambridge University Press. ISBN 978-0-521-33655-0. [2] https://www.sfu.ca/~renns/lbullets.html#bullets [3] https://en.wikipedia.org/wiki/Soliton#cite_note-2 [4] https://www.laserfocusworld.com/fiber-optics/article/16556409/solitons-balance-signals-for-transmission [5] http://www.scholarpedia.org/article/Soliton [6] G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, 2007) [7] https://wp.optics.arizona.edu/kkieu/wp-content/uploads/sites/29/2019/04/Solitons-in-optical-fibers-04-05-19.pdf [8] https://www.lightwaveonline.com/optical-tech/transport/article/16647155/future-of-solitons [9] Davydov, Aleksandr S. (1991). Solitons in molecular systems. Mathematics and its applications (Soviet Series). Vol. 61 (2nd ed.). Kluwer Academic Publishers. ISBN 978-0-7923-1029-7. [10] Yakushevich, Ludmila V. (2004). Nonlinear physics of DNA (2nd revised ed.). Wiley-VCH. ISBN 978-3-527-40417-9. [11] Sinkala, Z. (August 2006). \"Soliton/exciton transport in proteins\". J. Theor. Biol. 241 (4): 919–27. Bibcode:2006JThBi.241..919S. CiteSeerX 10.1.1.44.52. doi:10.1016/j.jtbi.2006.01.028. PMID 16516929. [12] Heimburg, T., Jackson, A.D. (12 July 2005). \"On soliton propagation in biomembranes and nerves\". Proc. Natl. Acad. Sci. U.S.A. 102 (2): 9790–5. Bibcode:2005PNAS..102.9790H. doi:10.1073/pnas.0503823102. PMC 1175000. PMID 15994235. [13] Heimburg, T., Jackson, A.D. (2007). \"On the action potential as a propagating density pulse and the role of anesthetics\". Biophys. Rev. Lett. 2: 57–78. arXiv:physics/0610117. Bibcode:2006physics..10117H. doi:10.1142/S179304800700043X. S2CID 1295386. [14] Andersen, S.S.L., Jackson, A.D., Heimburg, T. (2009). \"Towards a thermodynamic theory of nerve pulse propagation\". Prog. Neurobiol. 88 (2): 104–113. doi:10.1016/j.pneurobio.2009.03.002. PMID 19482227. S2CID 2218193.[dead link] [15] Hameroff, Stuart (1987). Ultimate Computing: Biomolecular Consciousness and Nanotechnology. Netherlands: Elsevier Science Publishers B.V. p. 18. ISBN 0-444-70283-0. [16] Weston, Astrid; Castanon, Eli G.; Enaldiev, Vladimir; Ferreira, Fábio; Bhattacharjee, Shubhadeep; Xu, Shuigang; Corte-León, Héctor; Wu, Zefei; Clark, Nicholas; Summerfield, Alex; Hashimoto, Teruo (April 2022). \"Interfacial ferroelectricity in marginally twisted 2D semiconductors\". Nature Nanotechnology. 17 (4): 390–395. arXiv:2108.06489. Bibcode:2022NatNa..17..390W. doi:10.1038/s41565-022-01072-w. ISSN 1748-3395. PMC 9018412. PMID 35210566. [17] Alden, Jonathan S.; Tsen, Adam W.; Huang, Pinshane Y.; Hovden, Robert; Brown, Lola; Park, Jiwoong; Muller, David A.; McEuen, Paul L. (2013-07-09). \"Strain solitons and topological defects in bilayer graphene\". Proceedings of the National Academy of Sciences. 110 (28): 11256–11260. arXiv:1304.7549. Bibcode:2013PNAS..11011256A. doi:10.1073/pnas.1309394110. ISSN 0027-8424. PMC 3710814. PMID 23798395. [18] Zhang, Shuai; Xu, Qiang; Hou, Yuan; Song, Aisheng; Ma, Yuan; Gao, Lei; Zhu, Mengzhen; Ma, Tianbao; Liu, Luqi; Feng, Xi-Qiao; Li, Qunyang (2022-04-21). \"Domino-like stacking order switching in twisted monolayer–multilayer graphene\". Nature Materials. 21 (6): 621–626. Bibcode:2022NatMa..21..621Z. doi:10.1038/s41563-022-01232-2. ISSN 1476-4660. PMID 35449221. S2CID 248303403. [19] Jiang, Lili; Wang, Sheng; Shi, Zhiwen; Jin, Chenhao; Utama, M. Iqbal Bakti; Zhao, Sihan; Shen, Yuen-Ron; Gao, Hong-Jun; Zhang, Guangyu; Wang, Feng (2018-01-22). \"Manipulation of domain-wall solitons in bi- and trilayer graphene\". Nature Nanotechnology. 13 (3): 204–208. Bibcode:2018NatNa..13..204J. doi:10.1038/s41565-017-0042-6. ISSN 1748-3387. PMID 29358639. S2CID 205567456. [20] Nam, Nguyen N. T.; Koshino, Mikito (2020-03-16). \"Erratum: Lattice relaxation and energy band modulation in twisted bilayer graphene [Phys. Rev. B 96 , 075311 (2017)]\". Physical Review B. 101 (9): 099901. Bibcode:2020PhRvB.101i9901N. doi:10.1103/physrevb.101.099901. ISSN 2469-9950. S2CID 216407866. [21] Dai, Shuyang; Xiang, Yang; Srolovitz, David J. (2016-08-22). \"Twisted Bilayer Graphene: Moiré with a Twist\". Nano Letters. 16 (9): 5923–5927. Bibcode:2016NanoL..16.5923D. doi:10.1021/acs.nanolett.6b02870. ISSN 1530-6984. PMID 27533089. [22] Kosevich, A. M.; Gann, V. V.; Zhukov, A. I.; Voronov, V. P. (1998). \"Magnetic soliton motion in a nonuniform magnetic field\". Journal of Experimental and Theoretical Physics. 87 (2): 401–407. Bibcode:1998JETP...87..401K. doi:10.1134/1.558674. S2CID 121609608. Archived from the original on 2018-05-04. Retrieved 2019-01-18. [23] Iwata, Yoritaka; Stevenson, Paul (2019). \"Conditional recovery of time-reversal symmetry in many nucleus systems\". New Journal of Physics. 21 (4): 043010. arXiv:1809.10461. Bibcode:2019NJPh...21d3010I. doi:10.1088/1367-2630/ab0e58. S2CID 55223766. [24] Lightwave, \"Handling Special Effects: Nonlinearity, Chromatic Dispersion, and Soliton Waves,\" July 2000, page 84.) [25] Lightwave, \"Handling Special Effects: Nonlinearity, Chromatic Dispersion, and Soliton Waves,\" July 2000, page 84.) [26] https://www.nakazawa.riec.tohoku.ac.jp/English/reserch/re01.html [27] https://www.intechopen.com/chapters/72536 [28] Youssoufa M, Dafounansou O, Mohamadou A. W-shaped, dark and grey solitary waves in the nonlinear Schrödinger equation competing dual power-law nonlinear terms and potentials modulated in time and space. Journal of Modern Optics. 2019;66(5):530-540 [29] https://en.wikipedia.org/wiki/Soliton_(optics) [30] https://physicsworld.com/a/dark-solitons-emerge-into-the-light/ [31] P. Emplit et al., Opt. Commun. 62, 374 (1987). [32] Y.S. Kivshar and S.K. Turitsyn, Opt. Lett. 18, 337 (1993); Y.S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998), and refs. Therein. [33] Y.S. Kivshar, Opt. Lett. 17, 1322 (1992); V.V. Afanasyev et al., Opt. Lett. 14, 805 (1989).

Copyright

Copyright © 2023 Muhammad Arif Bin Jalil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.