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Privacy Statement. Microsoft Solitaire Collection. Official Club. The constraints arise almost always because the differential equations must obey a set of boundary conditions , and the boundary has a nontrivial homotopy group , preserved by the differential equations.
Thus, the differential equation solutions can be classified into homotopy classes. No continuous transformation maps a solution in one homotopy class to another.
The solutions are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include the screw dislocation in a crystalline lattice , the Dirac string and the magnetic monopole in electromagnetism , the Skyrmion and the Wess—Zumino—Witten model in quantum field theory , the magnetic skyrmion in condensed matter physics, and cosmic strings and domain walls in cosmology.
In , John Scott Russell describes his wave of translation. I followed it on horseback, 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 , was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.
Scott Russell spent some time making practical and theoretical investigations of these waves. He built wave tanks at his home and noticed some key properties:.
Scott Russell's experimental work seemed at odds with Isaac Newton 's and Daniel Bernoulli 's theories of hydrodynamics.
George Biddell Airy and George Gabriel Stokes had difficulty accepting Scott Russell's experimental observations because they could not be explained by the existing water wave theories.
Their contemporaries spent some time attempting to extend the theory but it would take until the s before Joseph Boussinesq  and Lord Rayleigh published a theoretical treatment and solutions.
In Norman Zabusky of Bell Labs and Martin Kruskal of Princeton University first demonstrated soliton behavior in media subject to the Korteweg—de Vries equation KdV equation in a computational investigation using a finite difference approach.
They also showed how this behavior explained the puzzling earlier work of Fermi, Pasta, Ulam, and Tsingou. In , Gardner, Greene, Kruskal and Miura discovered an inverse scattering transform enabling analytical solution of the KdV equation.
Note that solitons are, by definition, unaltered in shape and speed by a collision with other solitons. Solitons are also studied in quantum mechanics, thanks to the fact that they could provide a new foundation of it through de Broglie 's unfinished program, known as "Double solution theory" or "Nonlinear wave mechanics".
This theory, developed by de Broglie in and revived in the s, is the natural continuation of his ideas developed between and , which extended the wave-particle duality introduced by Albert Einstein for the light quanta , to all the particles of matter.
In , researchers from Tel-Aviv university measured an accelerating surface gravity water wave soliton by using an external hydrodynamic linear potential.
They also managed to excite ballistic solitons and measure their corresponding phases. Much experimentation has been done using solitons in fiber optics applications.
Solitons in a fiber optic system are described by the Manakov equations. Solitons' inherent stability make long-distance transmission possible without the use of repeaters , and could potentially double transmission capacity as well.
The above impressive experiments have not translated to actual commercial soliton system deployments however, in either terrestrial or submarine systems, chiefly due to the Gordon—Haus GH jitter.
Consequently, the long-haul fiberoptic transmission soliton has remained a laboratory curiosity. Solitons may occur in proteins  and DNA.
A recently developed model in neuroscience proposes that signals, in the form of density waves, are conducted within neurons in the form of solitons.
In magnets, there also exist different types of solitons and other nonlinear waves. Atomic nuclei may exhibit solitonic behavior.
Such conditions are suggested to exist in the cores of some stars in which the nuclei would not react but pass through each other unchanged, retaining their soliton waves through a collision between nuclei.
The Skyrme Model is a model of nuclei in which each nucleus is considered to be a topologically stable soliton solution of a field theory with conserved baryon number.
The bound state of two solitons is known as a bion ,    or in systems where the bound state periodically oscillates, a breather.
In field theory bion usually refers to the solution of the Born—Infeld model. The name appears to have been coined by G. Gibbons in order to distinguish this solution from the conventional soliton, understood as a regular , finite-energy and usually stable solution of a differential equation describing some physical system.
However, the solution of the Born—Infeld model still carries a source in the form of a Dirac-delta function at the origin. As a consequence it displays a singularity in this point although the electric field is everywhere regular.
In some physical contexts for instance string theory this feature can be important, which motivated the introduction of a special name for this class of solitons.
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