A Brief Historical Overview and Mathematical Equations of Soliton

Abstract

A soliton is a strongly stable, nonlinear, self-reinforcing, localised wave packet that maintains its shape even after colliding with other similar localised wave packets and propagates freely at a constant speed. The medium\'s balanced cancellation of nonlinear and dispersive effects is responsible for its extraordinary stability. Dispersive effects are a characteristics of several systems where a wave frequency determines how fast it travels. Subsequently, it was shown that soliton solutions offer stable solutions for a broad range of weakly nonlinear dispersive partial differential equations that characterise physical systems. Solitons have a self-restoring quality that makes them desirable for long-distance, high-speed transmissions. The strain on networks is quickly reaching a critical threshold as transmission speeds over longer distances are now approaching 40 Gbit/sec. In the future, solitons might offer intriguing alternatives.

Introduction

I. INTRODUCTION

It is challenging to come up with a soliton definition that is widely accepted. Three characteristics are assigned to solitons by Drazin & Johnson (1989, p. 15) [1]. They have three characteristics: they are localised inside a region, they have a permanent form, and they can interact with other solitons. They also emerge from collisions unaltered, save for a phase shift.There are more formal definitions available, but they involve a lot of maths. Furthermore, several scientists refer to events that lack these three characteristics as soliton phenomena for instance, the 'light bullets' in nonlinear optics are sometimes referred to as solitons even though they lose energy during interaction).[2]Permanent and localised wave shapes can be created by the interaction of dispersion and nonlinearity. Think about a light pulse passing through glass. We might imagine that this pulse is made up of several frequencies of light. These various frequencies move at different speeds through glass, which exhibits dispersion; as a result, the pulse's form varies over time. Nevertheless, there is also the nonlinear Kerr effect, in which the strength or amplitude of the light determines a material's refractive index at a certain frequency. When a pulse is perfectly shaped, the dispersion effect is precisely cancelled by the Kerr effect, and the pulse's shape remains constant over time. The pulse is a soliton as a result [3]. Solitons solve a large number of absolutely solvable models, such as the sine-Gordon equation, the nonlinear Schrödinger equation, the Korteweg–de Vries equation, and the coupled nonlinear Schrödinger equation. The stability of the soliton solutions is attributed to the integrability of the field equations, which is commonly achieved using the inverse scattering transform. One large and active area of mathematical research is the mathematical theory of these equations.[3]

Certain tidal bore phenomena, which occur in a few rivers, such as the River Severn, are 'undular,' consisting of a wavefront followed by a train of solitons. Additional solitons arise from the propagation of seafloor topography-initiated underwater internal waves along the oceanic pycnocline. There are other atmospheric solitons as well. One example is the morning glory cloud in the Gulf of Carpentaria, which is created by massive linear roll clouds caused by pressure solitons moving across a temperature inversion layer. Pressure solitons are proposed as an explanation for the signal conduction within neurons in the recently developed, but not commonly accepted, soliton model of neuroscience [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. Rather than the field equations' integrability, topological constraints are the cause of soliton stability. The differential equations have to follow a set of boundary conditions, and the differential equations preserve the boundary's nontrivial homotopy group, which is why constraints almost always exist. The solutions to differential equations can therefore be categorised into homotopy classes [3].

In order to enhance the quality of signal transmission, soliton transmission in optical fibres performs a balancing act to counteract two main types of pulse degradation. The dispersion that occurs when pulses move over extended stretches of fibre 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.

The combined effect of the two effects usually exacerbates the situation, but for specific shapes and power levels of optical pulses, the effects can cancel each other out, at least to a first-order approximation. These kinds of pulses are referred to as solitons.Solitons' natural resilience for high-speed gearbox over extended distances is one of their strongest draws. Although soliton attenuation occurs, it is possible to make soliton intrinsically stable over extended fibre lengths. This provides a method to prevent signal quality degradation due to dispersion and nonlinear effects, which is a serious issue at 10 Gbit/s and becomes worse at higher transmission speeds. Because of these characteristics, researchers are working on soliton systems for upcoming 40-Gbit/s systems as well as long-haul 10-Gbit/s systems [4].

II. LITERATURE REVIEW

John Scott Russell (1808–1882) initially reported the soliton phenomenon in 1834 when he saw a lone wave in Scotland's Union Canal. He called it the "Wave of Translation" after he successfully replicated the phenomenon in a wave tank. The Korteweg–de Vries equation, which represents waves similar to those seen by Russell, has localised, strongly stable propagating solutions. Zabusky and Kruskal first used the name soliton to describe these solutions. With the "on" suffix, which was originally used to refer to particles like hadrons, baryons, and electrons and reflected their observed particle-like activity, the word was intended to describe the solitary nature of the waves.[3].

When a young engineer named John Scott Russell was engaged for a summer work in 1834 to look at ways to increase the effectiveness of designs for barges that were intended to traverse canals—particularly the Union Canal in Edinburgh, Scotland ,he made the first known observation of a lone wave. The barge abruptly stopped one August day when the tow rope holding the mules to it broke. However, the water mass in front of the blunt prow of the barge rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth, and well defined heap of water, which continued its course along the channel without change of form or diminution of speed." Russell (1844). Russell went on to investigate this accidental discovery and followed it the launched Wave of Translation  on horseback, and overtook it still rolling on at a rate of some eight or nine miles per hour, preserving its original form some thirty feet long and a foot to a foot and a half in height. Subsequently, he employed a wave tank in carefully regulated laboratory trials and published his findings in 1844 (Russell, 1844). He illustrated four points: He noticed single waves with a hyperbolic secant structure.When the initial mass of water is large enough, it can generate two or more independent, near-solitary waves that split over time.Waves in isolation can pass one another "without change of any kind".n a shallow water channel of height h , a solitary wave of amplitude A travels at a speed of [g(A+h)]1/2 , where g is the gravitational acceleration. That is, larger-amplitude waves move faster than smaller ones, a nonlinear effect. A nonlinear partial differential equation (PDE) was derived in 1895 by Diederick Korteweg, a Dutch physicist, and his pupil Gustav de Vries (KdV) (de Vries, 1895) (Scott, 2005), that is now named after them. Russell's experiments might 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 causes waves of different wavelengths to travel at different velocities, add together to determine the rate of change of the wave's height over time.

In addition to a solitary-wave solution that matched the wave that Russell had followed, Korteweg and de Vries also discovered a periodic solution. A compromise between dispersion and nonlinearity led to these solutions. Mathematicians, physicists, and engineers investigating water waves neglected their work and Russell's observations until 1965, when Norman Zabusky and Martin Kruskal published their numerical solutions of the KdV equation (and coined the word "soliton") (Zabusky, 1965). A continuous description of the oscillations of unidirectional waves propagating on the cubic, Fermi–Pasta–Ulam (FPU) nonlinear lattice (Fermi, 1955; Porter, 2009b; Weissert, 1997) was obtained by Kruskal from (1). Concurrently, Morikazu Toda made history by being the first to identify a soliton in a discrete, integrable system (now known as the Toda lattice) (Toda, 1967).[5]

Films demonstrating interacting solitary waves in an FPU lattice, the KdV equation, and a modified KdV equation were produced in 1965 by Gary Deem, Zabusky, and Kruskal (Deem, 1965); see the discussion in the review paper (Zabusky, 1984). We illustrate the dynamics of solitons in the space-time diagram of Figure 1 using the KdV equation. Robert Miura discovered a precise transition between this modified KdV equation and equation (1) after realising the significance of this result (Miura, 1976). This sparked interest in the mathematical study of solitons when Clifford Gardner, John Greene, Martin Kruskal, and Robert Miura used the inverse scattering method to solve the KdV equation's initial-value problem in 1967 (Miura, 1968; Gardner, 1967; Gardner, 1974). This gave rise to a suitable understanding of integrability for continuum frameworks. In 1972, establishing the integrability of the nonlinear Schrödinger (NLS) problem and the existence of soliton solutions, Vladimir Zakharov and Alexei Borisovich Shabat extended the inverse scattering approach. The sine-Gordon equation which was already known to be integrable based on Albert Backlünd's 19th century investigations of surfaces with constant negative Gaussian curvature was one of several nonlinear PDEs that Mark Ablowitz, David Kaup, Alan Newell, and Harvey Segur proved to have soliton solutions for in 1973. Subsequently, other researchers have created associated soliton solutions and deduced various integrable PDEs (in one and several spatial dimensions). What defines a "soliton" in several spatial dimensions needs to be more nuanced, as the Kadomtsev–Petviashvili (KP) equation shown. Analytical strategies for exploring solitary waves in nonintegrable equations usually depend on variational approximations, asymptotic analysis, and/or perturbative methods (Kivshar, 1989). (Scott, 2005) Fibre Bragg grating coupled mode equations are a notable illustration of a nonintegrable system with accurate solutions for isolated solitary waves in optics.One of the most active fields in mathematics and physics today is the study of solitons and solitary waves (Scott, 2005). It has influenced a wide range of disciplines, from experimental science to the purest mathematics. Important findings in integrable systems, nonlinear dynamics, biology, optics, supersymmetry, and other fields have resulted from this. We will talk about some of the varieties and uses of solitary waves later in this piece. [5]

III. MATHEMATICAL EQUATIONS OF  SOLITON

A non-linear evolution equation that is solved at each instant and localised in a bounded domain of space, with the domain's size being bounded in time and its centre of motion being understood as the motion of a particle. The Korteweg–de Vries equation's soliton solution [6]

Conclusion

Solitons are light pulses that have the ability to travel very long distances without broadening in a non-linear dispersive medium. They have therefore generated a lot of attention in the communications industry. The reason solitons function in optical fibres is due to two effects that, under the right circumstances, can balance each other out. One of them is chromatic dispersion, which occurs when pulses with a range of wavelengths spread out during their passage in an optical fibre. The other is known as self-phase modulation, or SPM, and it covers a wider variety of wavelengths in the pulse spectrum. Dispersion and SPM can balance each other out, allowing the pulse to maintain its form or dispersion once it reaches equilibrium in the fibre. SPM can also cause the pulse to compress or even broaden more severely. Attenuation, on the other hand, weakens the pulse and impedes its ability to retain its form over the fibre span. In order to retain pulse forms and balance attenuation, optical amplifiers have been designed. [9],[10].

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] https://encyclopediaofmath.o rg/wiki/Soliton [7] G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, 2007) [8] https://wp.optics.arizona.edu/kkieu/wp-content/uploads/sites/29/2019/04/Solitons-in-optical-fibers-04-05-19.pdf [9] https://www.lightwaveonline.com/optical-tech/transport/article/16647155/future-of-solitons [10] Lightwave, \"Handling Special Effects: Nonlinearity, Chromatic Dispersion, and Soliton Waves,\" July 2000, page 84.)

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.