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An Interesting Possible Connection
Between Quantum Theory
and Solitons

Quantum Theory

Max Plank in 1900 established that energy changes take place in discrete amounts, effectively packets, which he decided to call quanta. The minimum amount of such energy changes are proportional to a particular constant which the scientific community decided to call Panck's Constant. Its value in MKS units is 6.62607015×10−34 joule-seconds. Its units of energy-time are described as an action.

Atomic Theory

In 1913 Niels Bohr published a marvelously accurate derivation of the spectra of hydrogen and other hydrogen-like atoms based upon a solar system-like model and the assumption that the electron in such atoms has an angular momentum equal to Planck's Constant. This subsequently led to the fundamental proposition of quantum theory that atomic-level particles have a minimum momentum and and a minimum kinetic energy proportional to Planck's Constant.

This meant that zero motion was not a natural state for a particle. A zero motion state for the electrostatically charged particles of a material corresponding to zero absolute temperature can only be approached by constraining their motion in a magnetic field.

The inherent quantum theoretic motion of particles is definite but unexplained. There several questions to be asked concerning this element of quantum theory. the first is where does this minimum energy come from? The second is how is the minimum enforced? Answers to both of these questions will be given later.

Wave Motion of Fluids

In the 18th century mathematical physicists worked out the equation for waves in an ideal fluid and its solution was found to be sinusoidal. The equation was

uxx − v²utt = 0

where u is wave height and a subscripted variable stands for a partial derivative.

It all seemed to be very simple. Then in 1834 a famous engineer, John Scott Russell, observed a water wave created by a towed canal boat which was not sinusoidal. It was basically just a hump of water.

During most of the 19th century there was a dilemma. The type of wave observed by Russell was a common-place reality but the wave equation with only sinusoidal solutions appeared to be mathematically rigorous.

Then in 1895 two Dutch mathematicians, Diederik Johannes Korteweg and Gustav de Vries, published an analysis of fluid waves that took into account the nonlinearities of fluid surfaces. The nonlinear partial differential wave equation they derived had solutions which corresponded to the type of wave observed by Russell. The Korteweg-de Vries (KdV) equation may be expressed as

ut + ux + uux + uxxx = 0

The KdV equation has solutions of the form

u(x, t) = (β/2)sech²((√β/2)(x − βt))

The parameter β which determines the amplitude also determines the velocity β3/2/2.

These solutions have a wave-like profile but also have an inherent motion based upon a parameter of the solution.


In the 1950's when digital computers became widely available mathematicians began to use them to get numerical approximations of the solutions to nonlinear partial differential equations. Of course one of those equations was the Korteweg-de Vries equations. Furthermore they tried experiments. They started with initial conditions with two wave profiles with different parameters so they had different speeds. When the waves crashed into each other there was a period in which the solution was chaotic, but eventually the wave profiles emerged from the collision unchanged. Thus the wave solutions had particle-like characteristics as well as wave characteristics. The solutions to other nonlinear partial differential equations were found also found to have this characteristic. The solutions were called solitons.

Other nonlinear partial differential equations were found to have solutions which emerged from collisions modified in structure in ways analogous to what happens in collisions of physical particles. These solutions were called solitary waves.

The common characteristic of these solutions to nonlinear partial differential equations is that they have inherent motion. Zero motion is not an option.

The first solitons found were one dimensional. Soon two dimensional solitons were found but not three dimensional ones. However now three dimensional solitons are known but not the ones corresponding to subatomic particles.


If physical particles are soliton-like solutions to some nonlinear partial differential equation that is not yet known that would explain the quantum theoretic phenomenon of physical particles having a minimum level of momentum. When a physical particle is created there has to be enough energy to account not only for its mass but also for its quantum theoretic motion. It is analogous to an electron in an atom acquiring a specific amount of kinetic energy and a a specific amount of potential energy.

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