Soliton

from Wikipedia, the free encyclopedia

A soliton is a wave packet that moves through a dispersive and at the same time nonlinear medium without changing its shape . In a collision with similar wave packets it is not an interaction ; if, on the other hand, there is an interaction in which energy is exchanged, then it is a solitary wave .

In general, a wave packet contains, as can be shown with the aid of Fourier analysis , harmonic waves of several frequencies . If the speed of propagation in the medium is different at different frequencies, the packet changes its shape over time. This is called the dispersion of the phase velocity .

Nonlinear effects can now convert the individual frequencies that make up a wave packet into one another. If this happens in such a way that the faster frequency components are converted into slower ones and slower ones into faster ones, a dynamic equilibrium can develop: a dimensionally stable soliton.

Soliton in the wave channel in the laboratory - water wave

history

The soliton phenomenon was first described in 1834 by the young engineer John Scott Russell . Russell rode several miles next to a ten meter long and half a meter high water wave that was spreading in a narrow Scottish canal and observed that its waveform changed little.

He further researched the phenomenon using a tank in his workshop. In doing so, he discovered some key properties of these waves:

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

It was not until 1895 that the phenomenon could also be theoretically explained by the Korteweg-de-Vries equation , but the importance of the discovery was not recognized until the 1960s. In 1973 the existence of optical solitons in fiber optics was theoretically predicted and experimentally demonstrated for the first time in 1980.

application

In optical fibers , light pulses of low intensity are wave packets in a linear medium. They widen over time due to dispersion. When used for signal transmission, this deteriorates the signal quality because intersymbol interference can occur. As a result, the maximum possible transmission distance or the transmission rate is limited. A soliton, on the other hand, is a pulse of light that does not change as it propagates. This means that messages can theoretically be transmitted over any long distance, and with sufficiently short light pulses, a very high data transmission rate can be achieved.

In optical waveguides, solitons can be generated in the range of anomalous dispersion (the speed of propagation is higher here at higher frequencies) - that is, with conventional glass fibers at wavelengths of λ> 1.3 µm. Only a few milliwatts of power is required for this. The pulse duration is a few picoseconds , which enables transmission rates in the range of terabits / second (10 12  bit / s) over long distances. Attenuation and scattering losses exist in real media , which leads to a decrease in energy. This destroys the balance between dispersion and non-linearity, so that the soliton dissolves. In real data transmission systems the solitons have to be amplified again and again (about every 20 km).

In tests in special fiberglass rings, solitons have already been transmitted over 180 million kilometers without noticeable pulse broadening. With lasers, mode coupling can be used to generate solitons, which are a prerequisite for operating a frequency comb . In this case, even after hours, no deliquescence of a stored pulse is observed.

In addition to the usual spin waves , soliton-like excitations also exist in low-dimensional magnets . They have long been extensively studied both theoretically and experimentally.

More examples of solitons

Soliton equations from mathematical physics

The following equations are some examples of equations in mathematical physics with soliton solutions:

There are a few other examples, such as the modified Korteweg-de-Vries equation, as well as entire hierarchies of equations derived from them. They can often be solved exactly using the inverse scattering transformation method . Further solution methods are the direct method of Ryōgo Hirota and Bäcklund transformations .

One-dimensional (1D) FDTD simulation of a soliton with force

The following video shows an FDTD simulation of two solitons at rest according to the Sinus-Gordon equation (see below). Both also send out pressure-speed fields with different polarity. Because the ends of the room are not properly terminated, reflections also occur.

The simulation will be explained below.

Sinus Gordon equation

The Sinus Gordon equation is a partial differential equation (DGL) of the second order and reads:

(The sub-index denotes the partial derivative according to the relevant variable.)

Its name was not taken very seriously from the fact that it has the form of a Klein-Gordon equation , in which the mass is "replaced" by the sine function (the form of the Klein-Gordon equation also results as the first term of Series expansion of the sine). It describes the relationship between time , position and excitation in a one-dimensional space. The situation can be illustrated as a chain of transversely oscillating pendulums, with springs between the pendulums, which couple the rotation angle of the pendulums to one another. In the example, the location coordinate is also given by the number of pendulums and the elapsed time. DGL solutions include two oppositely structured solitons, known as kink and antikink. For small angles the equation also describes advancing waves. A soliton can rest or move at a constant speed. It is characterized in that the angle differences within the soliton result in a full circle. The soliton has a fixed size and can only exist as a whole. It is insensitive to disturbances of a single pendulum, it then reacts elastically.

Discrete Time and Space Simulation - FDTD

The finite difference method in the time domain (FDTD) is mainly used to simulate the propagation of electromagnetic fields in three-dimensional space. The room is divided into a rectangular grid made of tiny cubes. The behavior of the fields is simulated by close effects between direct neighboring cubes with difference equations. The calculations are often carried out by graphics processors (GPUs), as the type of algorithms or arithmetic operations used can usually be implemented more effectively and are more powerful in this area than on conventional main processors (CPU). In pseudocode, two calculation steps executed cyclically one after the other:

Step 1 

SELF(VAR1) += RIGHT(VAR2) - LEFT(VAR2)

step 2 

SELF(VAR2) += RIGHT(VAR1) - LEFT(VAR1)

This calculation rule uses a kind of double buffer technique and enables the propagation of sinusoidal waves.

Generation of the above video

The video provides a 1D FDTD space with the Sinus Gordon equation between the space points ( iterator ). In addition, a second neighborhood relationship is implemented which enables pressure-velocity waves.

Both relationships are coupled with each other (Soliton transmits and is added to the angular velocity).

You can see each other in the video:

At the start, first two opposing solitons (recognizable as white bars in the first line, with the opposite course of the sine component in the second line) are initialized ( generator ). Since this is not a stationary solution of Eq. (1), superfluous energy is sent out as a wave. Then the solitons send a p / v field (see fourth and fifth lines). When the p / v field reaches the other soliton, the solitons start moving. Because, for the sake of simplicity, the ends of the room are not perfectly terminated, waves are reflected there. Eventually the solitons meet in the middle and are destroyed by annihilation . What remains is the energy in the form of waves.

Purpose of the video

It models the issues of particles , forces and waves in space and their relationship. Thus, special solitons are a possibility of particles that already show several properties: antiparticles, movement, energy, elasticity, size and continuity.

literature

  • Alan C. Newell : Solitons in Mathematics and Physics . SIAM 1985
  • Mark J. Ablowitz , PA Clarkson: Solitons, Nonlinear Evolution Equations and Inverse Scattering . Cambridge University Press, 1991
  • George L. Lamb: Elements of Soliton Theory . Wiley, 1980
  • AC Scott, FYF Chu, DW McLaughlin: Soliton: A new concept in applied science. In: Proceedings of the IEEE , 61, 1973, No. 10, pp. 1443-1482.
  • Hans-Jürgen Mikeska , M. Steiner: Solitary excitations in one-dimensional magnets. In: Advances in Physics , 40, 1991, No. 3, pp. 191-356, doi: 10.1080 / 00018739100101492 .
  • AV Buryak, P. Di Trapani, DV Skryabin, S. Trillo: Optical solitons due to quadratic nonlinearities: From basic physics to futuristics applications. In: Physics Reports , 370, 2002, No. 2, pp. 63-235.
  • Reinhard Meinel , Gernot Neugebauer , Heinz Steudel: Solitons - Nonlinear Structures. Akademie Verlag, Berlin 1991, ISBN 3-05-500710-7 .
  • Philip G. Drazin et al .: Solitons - an introduction. Cambridge Univ. Press, Cambridge 2002, ISBN 0-521-33389-X .
  • Thierry Dauxois, Michel Peyrard: Physics of solitons. Cambridge Univ. Press, Cambridge 2006, ISBN 0-521-85421-0 .
  • R. Rajaraman: Solitons and instantons - an introduction to solitons and instantons in quantum field theory. Elsevier, Amsterdam 2005, ISBN 0-444-87047-4 .

Web links

Individual evidence

  1. ↑ Thomas Udem: The measurement of the frequency of light with mode-locked lasers . Habilitation thesis. 2002, p. 16 ( mpg.de [PDF; accessed on February 27, 2018]).
  2. ↑ Markus Dietrich, Hans-Josef Patt: Wave machine for demonstrating and measuring harmonic and anharmonic wave phenomena (solitons) , uni-saarland.de (PDF)