SolitonsSolitons are stable solutions to a broad class of nonlinear difference and differential equations, with an enormous practical impact. Solitons are among paradigm structural selforganizers in Nature, that describe a wide range of phenomena in physical, chemical and biological scenarios. The effective analysis of solitons and their properties necessitates mastering a large expanse of approaches from analytical, topological and geometrical techniques to computational and numerical methods and modern visualization tools. Efforts to understand the theoretical structure and properties of solitons continue to contribute significantly to developments for example in numerical analysis, symbolic computer algebra and differential geometry, and even to the development of entirely new kind of supercomputers. The ongoing advancements in computers, in turn, facilitates the explicit construction and detailed investigation of increasingly many new kind of complex solitons. The aim is to search for novel interdisciplinary links between the powerful techniques of quantum field and string theories that were originally developed to address issues in elementary particle physics and gravity theory, and emerging problems in biology and cosmology where entirely new kind of solitons appear to play a pivotal role. These sophisticated techniques of theoretical and mathematical physics have already been extensively utilized to address problems in condensed matter physics, to the extent that the interface between high energy, gravity and condensed matter is now among the most fruitful domains of ground breaking new ideas.
They occur in a number of systems:
Integrable SystemsThey correspond to a class of nonlinear partial differential equations knows as soliton equations. The key of these equations is that they possess special types of elementary solutions (taking the form of localized disturbances, or pulses), which retain their shape after interactions among themselves (at least in the integrable systems), and thus they act like particles. These localized disturbances have come to be known as solitons. In integrable systems the highly non-trivial equations of motion can be written as the compatibility condition of a linear system of partial differential equations, known as Lax pair and the solutions can be obtained analytically through many approaches.
Topological SystemsThey admit topological solitons, the stability of which depends on nontrivial topology. These solutions can be studied using topological methods. Moreover, they may be assigned an integer valued topological charge, which is conserved, and has a natural physical interpretation. One can think of the solitons as subatomic particles, and the topological charges as one of the conserved quantities of particle physics.
Some known examples are: Monopoles, Skyrmions, Instantons and their variants that is when coupled with gravity (cosmology) or with fermions (supersymmetric theories).
Discrete ModelsThe existence of lattice versions of either topological or integrable models which preserve their topological character or integrability, respectively, is an interesting problem. For computational and other reasons, it is useful to find discrete versions of the systems in which the topological stability or the existence of the Lax pair is maintained.
Models from Biology to CosmologySolitons are ubiquitous and widely studied objects that can be materialized in a wide range of practical and theoretical scenarios. For example, solitons can be deployed for data transmission in transoceanic cables, for conducting electricity in organic polymers and for describing chemical energy transportation in proteins. Solitons explain the formation of the morning glory cloud in atmosphere, the Meissner effect in superconductivity and dislocations in liquid crystals. Solitons model hadronic particles, magnetic monopoles and cosmic strings in nuclear physics, high energy physics and cosmology.