Солитоны: самоподдерживающиеся волновые пакеты.

Self reinforcing single wave packet

In mathematics and physics, a soliton is a nonlinear, self reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a balanced cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". The term soliton was coined by Zabusky and Kruskal to describe localized, strongly stable propagating solutions to the Korteweg–de Vries equation, which models waves of the type seen by Russell. The name was meant to characterize the solitary nature of the waves, with the 'on' suffix recalling the usage for particles such as electrons, baryons or hadrons, reflecting their observed particle like behaviour.

Dispersion and nonlinearity can interact to produce permanent and localized wave forms. Consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of several different frequencies. Since glass shows dispersion, these different frequencies travel at different speeds and the shape of the pulse therefore changes over time. However, also the nonlinear Kerr effect occurs; the refractive index of a material at a given frequency depends on the light's amplitude or strength. If the pulse has just the right shape, the Kerr effect exactly cancels the dispersion effect and the pulse's shape does not change over time. Thus, the pulse is a soliton. See soliton (optics) for a more detailed description. Many exactly solvable models have soliton solutions, including the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and the sine Gordon equation. The soliton solutions are typically obtained by means of the inverse scattering transform, and owe their stability to the integrability of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research. Some types of tidal bore, a wave phenomenon of a few rivers including the River Severn, are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the undersea internal waves, initiated by seabed topography, that propagate on the oceanic pycnocline. Atmospheric solitons also exist, such as the morning glory cloud of the Gulf of Carpentaria, where pressure solitons traveling in a temperature inversion layer produce vast linear roll clouds. The recent and not widely accepted soliton model in neuroscience proposes to explain the signal conduction within neurons as pressure solitons. A topological soliton, also called a topological defect, is any solution of a set of partial differential equations that is stable against decay to the "trivial solution". Soliton stability is due to topological constraints, rather than integrability of the field equations. 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.

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. Year Discovery 1973 Akira Hasegawa of AT&T Bell Labs was the first to suggest that solitons could exist in optical fibers, due to a balance between self phase modulation and anomalous dispersion. Also in 1973 Robin Bullough made the first mathematical report of the existence of optical solitons. He also proposed the idea of a soliton based transmission system to increase performance of optical telecommunications. 1987  – from the Universities of Brussels and Limoges – made the first experimental observation of the propagation of a dark soliton, in an optical fiber. 1988 Linn F. Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called the Raman effect, named after Sir C. V. Raman who first described it in the 1920s, to provide optical gain in the fiber. 1991 A Bell Labs research team transmitted solitons error free at 2.5 gigabits per second over more than 14,000 kilometers, using erbium optical fiber amplifiers (spliced in segments of optical fiber containing the rare earth element erbium). Pump lasers, coupled to the optical amplifiers, activate the erbium, which energizes the light pulses. 1998 Thierry Georges and his team at France Telecom R&D Center, combining optical solitons of different wavelengths (wavelength division multiplexing), demonstrated a composite data transmission of 1 terabit per second (1,000,000,000,000 units of information per second), not to be confused with Terabit Ethernet. 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. The GH jitter requires sophisticated, expensive compensatory solutions that ultimately makes dense wavelength division multiplexing (DWDM) soliton transmission in the field unattractive, compared to the conventional non return to zero/return to zero paradigm. Further, the likely future adoption of the more spectrally efficient phase shift keyed/QAM formats makes soliton transmission even less viable, due to the Gordon–Mollenauer effect. Consequently, the long haul fiberoptic transmission soliton has remained a laboratory curiosity. 2000 Steven Cundiff predicted the existence of a vector soliton in a birefringence fiber cavity passively mode locking through a semiconductor saturable absorber mirror (SESAM). The polarization state of such a vector soliton could either be rotating or locked depending on the cavity parameters. 2008 D. Y. Tang et al. observed a novel form of higher order vector soliton from the perspectives of experiments and numerical simulations. Different types of vector solitons and the polarization state of vector solitons have been investigated by his group.

Solitons may occur in proteins and DNA. Solitons are related to the low frequency collective motion 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. Solitons can be described as almost lossless energy transfer in biomolecular chains or lattices as wave like propagations of coupled conformational and electronic disturbances.

Solitons can occur in materials, such as ferroelectrics, in the form of domain walls. Ferroelectric materials exhibit spontaneous polarization, or electric dipoles, which are coupled to configurations of the material structure. Domains of oppositely poled polarizations can be present within a single material as the structural configurations corresponding to opposing polarizations are equally favorable with no presence of external forces. The domain boundaries, or “walls”, that separate these local structural configurations are regions of lattice dislocations. The domain walls can propagate as the polarizations, and thus, the local structural configurations can switch within a domain with applied forces such as electric bias or mechanical stress. Consequently, the domain walls can be described as solitons, discrete regions of dislocations that are able to slip or propagate and maintain their shape in width and length. In recent literature, ferroelectricity has been observed in twisted bilayers of van der Waal materials such as molybdenum disulfide and graphene. The moiré superlattice that arises from the relative twist angle between the van der Waal monolayers generates regions of different stacking orders of the atoms within the layers. These regions exhibit inversion symmetry breaking structural configurations that enable ferroelectricity at the interface of these monolayers. The domain walls that separate these regions are composed of partial dislocations where different types of stresses, and thus, strains are experienced by the lattice. It has been observed that soliton or domain wall propagation across a moderate length of the sample (order of nanometers to micrometers) can be initiated with applied stress from an AFM tip on a fixed region. The soliton propagation carries the mechanical perturbation with little loss in energy across the material, which enables domain switching in a domino like fashion. These magnetic solitons are an exact solution of classical nonlinear differential equations — magnetic equations, e. g. the Landau–Lifshitz equation, continuum Heisenberg model, Ishimori equation, nonlinear Schrödinger equation and others.

Atomic nuclei may exhibit solitonic behavior. Here the whole nuclear wave function is predicted to exist as a soliton under certain conditions of temperature and energy. 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. The interference type forces between solitons could be used in making bions. However, these forces are very sensitive to their relative phases. Alternatively, the bound state of solitons could be formed by dressing atoms with highly excited Rydberg levels. The word regular means a smooth solution carrying no sources at all. 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. On the other hand, when gravity is added (i. e. when considering the coupling of the Born–Infeld model to general relativity) the corresponding solution is called EBIon, where "E" stands for Einstein.

Erik Lentz, a physicist at the University of Göttingen, has theorized that solitons could allow for the generation of Alcubierre warp bubbles in spacetime without the need for exotic matter, i. e., matter with negative mass.