In physics and mathematics, a soliton is a physical solitary wave (in a liquid, gas, plasma, or solid) that propagates non-dispersively with a negligible, or small, loss of energy and can retain its shape and speed after colliding with another soliton. A soliton can be also be defined as a mathematical model of such a physical solitary wave. Mathematical solitons are solutions of a special class of non-linear partial differential equations.
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- ...attributes may be maintained because of deformations in fields. Such conservation laws are called topological. Thus, it may happen that a knot in a set of field lines, called a soliton, cannot be smoothed out. As a result, the soliton is prevented from dissipating and behaves much like a particle. A classic example is a magnetic monopole, which has not been found in nature but shows up as twisted configurations in some field theories.
In the traditional view, then, particles such as electrons and quarks (which carry Noether charges) are seen as fundamental, whereas particles such as magnetic monopoles (which carry topological charge) are derivative. In 1977, however, Claus Montonen, now at the Helsinki Institute of Physics in Finland, and David I. Olive, now at the University of Wales at Swansea, made a bold conjecture. Might there exist an alternative formulation of physics in which the roles of Noether charges (like electrical charge) and topological charges (like magnetic charge) are reversed? In such a “dual” picture, the magnetic monopoles would be the elementary objects, whereas the familiar particles—quarks, electrons and so on—would arise as solitons.
- A method is proposed to calculate quantum numbers on solitons in quantum field theory. The method is checked on previously known examples and, in a special model, by other methods. It is found, for example, that the fermion number on kinks in one dimension or on magnetic monopoles in three dimensions is, in general, a transcendental function of the coupling constant of the theories.
- While J. Scott Russell first observed solitons in water waves in Augst 1834, a full-fledged theory of solitons has only come of age in the last decade. This advance is due primarily in the discovery of a generalization of the Fourier transform, the inverse scattering transform. While this method can be used to solve exactly only a certain number of nonlinear equations, many of these are relevant to broad areas in physics.
- Chia-Hsiung Tze, (1982). "book review of 3 books: Elements of Soliton Theory by G. L. Lamb Jr.; Solitons: Mathematical Methods for Physicists by G. Eilenberger; and Solitons and the Inverse Scattering Transform by M. Ablowitz and H. Segur". Physics Today 35 (6): 55–56. DOI:10.1063/1.2915133.
- The idea that in some sense the ordinary proton and neutron might be solitons in a non-linear sigma model has a long history. The first suggestion was made by Skyrme more than twenty years ago ... David Finkelstein and Rubinstein showed that such objects could in principle be fermions ... in a paper that probably represented the first use of what would now be θ vacua in quantum field theory. A gauge invariant version was attempted by Faddeev ... Some relevant miracles are known to occur in two space-time dimensions ... ; there also exists a different mechanism by which solitons can be fermions ...