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The Nature of Fundamental Particles
as Solitons and Solitary Waves

The Wave-Particle Duality of Light

Around 1700 the scientific world, under the influence of Isaac Newton, believed that light consisted of particles. That remained the prevailing opinion until around 1800 when scientists such as Thomas Young created experiments that demonstrated the wave nature of light. In the latter part of the 19th century James Clerk Maxwell demonstrated that light was an oscillation of electromagnetic fields the issue seemed settled. Then shortly after 1900 Max Planck demonstrated that the frequency distribution of black body radiation could be explained if radiation energy is only transferred in discrete amounts, called quanta. Albert Einstein used this notion to explain the photoelectric effect. The discrete quanta later came to be known as photons.

Later Einstein remarked,

"All these 50 years of conscious brooding have brought me no nearer to the answer to the question 'What are light quanta?'. Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken."

The Wave-Particle Duality of Electrons
and Other Material Particles

The situation became more interesting in the early 1920's when the conjecture by Louis de Broglie of a wave aspect to particles was confirmed by experiments. De Broglie speculated that if a particle of mass m and velocity v had a wave aspect the characteristics of the wave would be

the de Broglie wavelength
λ = h/p

where h is Planck's constant and p is the momentum of the particle. Ignoring relativistic corrections p is equal to mv.

The frequency ν is given by

ν = c/λ

where c is the speed of light in a vacuum.

Bohr's Principle of Complementary

In the late 1920's Neils Bohr formulated his resolution of the issue of the wave-particle duality of particles. Instead of saying that electrons, photons and so forth sometimes behave like particles and sometimes like waves, he said that exists some structure that has both wave and particle characteristics. He called this the Principle of Complementarity. He recognized that the wave-particle dichotomy was false.

There are in fact general structures that would satisfy Bohr's Principle of Complementarity. They are called solitions and solitary waves. The story of the formulation of the concept of solitions is one of the most fascinating in mathematical physics.

The Development of the Soliton Concept

In 1834 the noted engineer John Scott Russell was riding a horse along a canal near Edinburgh in Scotland. He noticed that a mound of water was being pushed along with a boat. The boat stopped and the single peaked wave continued on. Russell rode along with the wave and followed it through the canal system. Here is his exact description.

I believe I shall best introduce this phenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped -- no so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, 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 dimunition of speed. I followed it on horse back, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now very generally bears.

When he told others of his observation some of them said that it could not be. They believed there was only only one particular equation for water waves and its solution was a sinusoidal function extending to infinity in both directions.

Korteweg and de Vries

The issue of the explanation of Russell's wave remained unresolved until 1895. In that year two Dutch mathematical physicists published an article in which they showed that surface waves on a liquid should satisfy a particular nonlinear partial differential equation.

The equation which came to be known as the Korteweg-de-Vries (KdV) equation is of the following form

ut = uxxx + 6uux

where u is the vertical displacement from equilibrium, t is time and x is horizontal distance. The subscripts denote partial derivatives.

This equation has a solution of the nature of the one witnessed by John Scott Russell.

u(x, t) = 2a²·sech²(a(x + 4a²t))

where sech(z) is the hyperbolic secant function and is equal to 2/(exp(z)+exp(−z)). Its square has the shape shown below

The parameter a determines not only the amplitude of the wave (2a²) but also the velocity of the wave (4a³). Thus the larger amplitude wave travels at a faster speed than a lower amplitude wave. For a positive value of a the wave travels to the left and for a negative value to the right. For more on the sech solution see Sech².

The KdV equation resolved the issue of Russell's wave but was otherwise of little significance.

The Discovery of Solitons

In the 1950's when mathematicians began to have access to computers they used them to solve equations. They applied the techniques they were developing to any equation they found. When applied to the KdV equation they found a form that moved as a unit.

Around 1965 M.D. Kruskal and N.J. Zabusky discovered a remarkable phenomenon. If a solution with a lower value of a is initially situated to the left of a solution with higher value of a the two solutions will crash into each other. In the interval of the crash the solution will fluctuate chaotically but subsequently the wave forms will reappear unaffected by the encounter. In other words, the wave forms behave as particles.

Zabusky and Kruskal coined the term soliton to describe the wave forms. However that naming is a bit misleading. The wave forms have no existence separate from the nonlinear partial differential equation for which they are solutions. It is the nonlinear partial differential equation that it the source of the phenomenon.

The research into the phenomenon found that there are conservation laws that are satisfied. Kruskal and Zabusky found two such conserved quantities and then their colleague R.M. Miura found another six. It was subsequently found that there are an infinite number of conserved quantities for the solition phenomenon for the KdV equation.

The research then explored other nonlinear partial differential equations for evidence of soliton phenomena. At first this exploration focused on minor modifications of the KdV equation. The Regularized Long Wave Equation (RLWE) was formulated by Peregrine (1966) as an alternative to KdV equation for studying soliton phenomenon. It was proposed because it would not have the same limitations for the size of the time step in numerical solution that the KdV has.

This modification of the KdV equation resulted in new phenomena. After collision of wave forms for the RLWE the wave forms reappeared but not quite of the same amplitude as the pre-collision forms. At first it was thought the descreptancy was due simply to numerical inaccuracies, but later it was established that changes occurred in the wave forms as a result of the collision. The term soliton was reserved for the cases in which the wave forms were exactly preserved. The soliton-like solutions to the RLWE were changed from their pre-collision form by the interaction of the collision. The name given to such wave forms was solitary waves. For more material on the solutions to the RLWE see RLWE and

Reference:

Roger Dodd, J. C. Eilbeck, John D. Gibbon and Hedley C. Morris, Solitions and Nonlinear Wave Equations, Academic Press, London, 1982.

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