In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.
Martin David Kruskal was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis. His most celebrated contribution was in the theory of solitons.
In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space.
Robin K. Bullough was a British mathematical physicist known for his contributions to the theory of solitons, in particular for his role in the development of the theory of the optical soliton, now commonly used, for example, in the theory of trans-oceanic optical fibre communication theory, but first recognised in Bullough's work on ultra-short optical pulses. He is also known for deriving exact solutions to the nonlinear equations describing these solitons and for associated work on integrable systems, infinite-dimensional Hamiltonian systems, and the statistical mechanics for these systems. Bullough also contributed to nonlinear mathematical physics, including Bose–Einstein condensation in magnetic traps.
In mathematical physics, the Degasperis–Procesi equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincaré conjecture and the Calabi conjecture. They are difficult to study: almost no general techniques exist that work for all such equations, and usually each individual equation has to be studied as a separate problem.
In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation
Charles Rogers Doering was a professor of mathematics at the University of Michigan, Ann Arbor. He is notable for his research that is generally focused on the analysis of stochastic dynamical systems arising in biology, chemistry and physics, to systems of nonlinear partial differential equations. Recently he had been focusing on fundamental questions in fluid dynamics as part of the $1M Clay Institute millennium challenge concerning the regularity of solutions to the equations of fluid dynamics. With J. D. Gibbon, he notably co-authored the book Applied Analysis of the Navier-Stokes Equations, published by Cambridge University Press. He died on May 15, 2021.
Stephen Ray Wiggins is an American, British and Cherokee applied mathematician best known for his contributions in nonlinear dynamics, chaos theory and nonlinear phenomena, with applications to Lagrangian aspects of fluid transport and mixing and phase space aspects of theoretical chemistry.
Norman J. Zabusky was an American physicist, who is noted for the discovery of the soliton in the Korteweg–de Vries equation, in work completed with Martin Kruskal. This result early in his career was followed by an extensive body of work in computational fluid dynamics, which led him in the latter years of his career to an examination of the importance of visualization in this field. In fact, he coined the term visiometrics to describe the process of using computer-aided visualization to guide one towards quantitative results.
George M. Zaslavsky was a Soviet mathematical physicist and one of the founders of the physics of dynamical chaos.
Institute of Mathematics of the National Academy of Sciences of Ukraine is a government-owned research institute in Ukraine that carries out basic research and trains highly qualified professionals in the field of mathematics. It was founded on 13 February 1934.
Geometric mechanics is a branch of mathematics applying particular geometric methods to many areas of mechanics, from mechanics of particles and rigid bodies to fluid mechanics to control theory.
Muthusamy Lakshmanan is an Indian theoretical physicist and a Ramanna fellow of the Department of Science and Technology at the Centre for Nonlinear Dynamics of Bharathidasan University. He has held several research fellowships which included Raja Rammanna fellowship of Department of Atomic Energy, Alexander von Humboldt fellowship, Japan Society for the Promotion of Science fellowship, Royal Society Nuffield Foundation fellowship, and NASI-Senior Scientist Platinum Jubilee Fellowship.
Igor Dmitrievich Chueshov was a Ukrainian mathematician. He was both a correspondent member of the Mathematics section of the National Academy of Sciences of Ukraine and a professor in the Department of Mathematical Physics and Computational Mathematics at the National University of Kharkiv.
Edriss Saleh Titi is an Arab-Israeli mathematician. He is Professor of Nonlinear Mathematical Science at the University of Cambridge. He also holds the Arthur Owen Professorship of Mathematics at Texas A&M University, and serves as Professor of Computer Science and Applied Mathematics at the Weizmann Institute of Science and Professor Emeritus at the University of California, Irvine.
Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.
Adrian Constantin is a Romanian-Austrian mathematician who does research in the field of nonlinear partial differential equations. He a professor at the University of Vienna and has made groundbreaking contributions to the mathematics of wave propagation. He is listed as an ISI Highly Cited Researcher with more than 160 publications and 11000 citations.
The Schamel equation (S-equation) is a nonlinear partial differential equation of first order in time and third order in space. Similar to a Korteweg de Vries equation (KdV), it describes the development of a localized, coherent wave structure that propagates in a nonlinear dispersive medium. It was first derived in 1973 by Hans Schamel to describe the effects of electron trapping in the trough of the potential of a solitary electrostatic wave structure travelling with ion acoustic speed in a two-component plasma. It now applies to various localized pulse dynamics such as:
