Soliton

A soliton is a wave packet moving through a dispersive and nonlinear medium at the same time and propagates without changing its shape. If, in a collision with identical wave packets, an interaction is replaced with energy, this is, this is a solitary wave. Shall be no exchange of energy, so it is a soliton.

A wave packet is, as can be shown by Fourier analysis of several frequencies. Is the propagation velocity in the medium at different frequencies are different, the packet having the time width. This is also called the group velocity dispersion. Nonlinear effects can now choose the individual frequencies that make up a wave packet is, converted into each other. If this is done such that the faster frequency components are converted into slower and slower to faster, then a dynamic equilibrium can form: a soliton.

History

The phenomenon of solitons were first described in 1834 by the young engineer John Scott Russell. Russell rode several miles in addition to about 10 feet long and about half a meter high water wave, which spread in a narrow Scottish channel, and observed that the waveform changed only slightly.

He studied the phenomenon further with the help of a tank in his workshop. He discovered a few key properties of these waves:

• The waves can continue stable over long distances.
• The velocity of the waves depends on the size of the shaft and the depth of water.
• Unlike normal waves they do not unite. A small wave is overtaken by a larger one.
• If a shaft is too big for the water depth, it divides into two waves: one large and one small.

It took until 1895 before the phenomenon could also be explained theoretically by the Korteweg -de Vries equation, however, until the 1960s the importance of the discovery was recognized. 1973, the existence of optical solitons was theoretically predicted in optical waveguides and demonstrated experimentally for the first time in 1980.

Properties

Are a necessary condition for the existence of solitons

• A non-linear wave equation and
• Dispersion.

Since there are an unlimited number of non-linear equations, there are as many types of solitons.

The most important non- linear differential equation associated with solitons is the Gross- Pitaevskii equation, which is a non-linear generalization of the Schrödinger equation. The Gross- Pitaevskii equation is used for example for the description of ultra- cold atoms and molecules or Bose -Einstein condensates. The non-linearity of the large - Pitaevskii equation is based on the consideration of interactions. The type of interaction is determined by a factor; for it is attractive for repulsive. One therefore distinguishes between bright solitons in attracting media and dark solitons in repulsive media.

Application

Low -intensity light pulses in the optical waveguides are wave packets in a nonlinear medium. They become wider due to the dispersion over time. Thus, the signal quality deteriorates because it can lead to intersymbol interference. As a result, the maximum transfer distance and the transmission rate is limited. A soliton is an optical pulse on the other hand, which does not change during propagation. This is theoretically a message transmission over arbitrarily long distances possible at sufficiently short light pulses, very high data transfer rate can be achieved.

In light waveguides to solitons can be in the range of anomalous dispersion of the group velocity ( the velocity of propagation is here at higher frequencies greater ) produce - so for silica fiber at wavelengths of λ > 1.3 microns. To this end, only a few milliwatts of power is required. The pulse duration is a few picoseconds, which (1012 bit / s) transmission rates in the range of terabits / second over long distances. In real media attenuation and scattering losses, which leads to a decrease in energy exist. This destroys the balance between dispersion and nonlinearity, so that dissolves the soliton. In real data transmission systems to the solitons must therefore (approximately every 20 km) nachverstärken over again.

In experiments in fiber rings solitons have been transmitted more than 180 million kilometers without any noticeable pulse broadening. Lasers can be generated by mode coupling solitons, which are a prerequisite for the operation of a frequency comb. This is observed even after hours, no deliquescence of a once stored pulse.

Solitonartige suggestions there are, in addition to the usual spin waves, even in low-dimensional magnets. They have long been studied both theoretically and experimentally in detail.

• Pororoca ( solitons on the Amazon )
• Morning Glory Cloud
• Impulse conduction in nerve cells
• Skyrmions

Soliton Equations of mathematical physics

The following equations are some examples of equations of mathematical physics:

• Korteweg -de Vries equation
• Nonlinear Schrödinger equation
• Toda chain

There are a few more examples of how the modified Korteweg -de Vries equation, as well as all the hierarchies of equations that are derived from these.