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Toda Lattice
The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system. It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian :\begin H(p,q) &= \sum_ \left(\frac +V(q(n+1,t)-q(n,t))\right) \end and the equations of motion :\begin \frac p(n,t) &= -\frac = e^ - e^, \\ \frac q(n,t) &= \frac = p(n,t), \end where q(n,t) is the displacement of the n-th particle from its equilibrium position, and p(n,t) is its momentum (mass m=1), and the Toda potential V(r)=e^+r-1. Soliton solutions Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way. The general N-soliton solution of the equation is : \begin q_N(n,t)=q_+ + \log \frac , \end where :C_N(n,t)=\Bigg(\frac\Bigg)_, with :\gamma_j(n,t)=\gamma_j\,e^ where \kappa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid State Physics
Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms a theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors. Background Solid materials are formed from densely packed atoms, which interact intensely. These interactions produce the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed, the atoms may be arranged in a regular, geometric pattern (crystalline solids, which include metals and ordinary water ice) or irregularly (an amorphous solid such as com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Completely Integrable System
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. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton
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. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread 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". Definition A single, consensus definition of a soliton is difficult to find. ascribe three properties to solitons: # They are of permanent form; # They are localized within a region; # They can interact with other solitons, and emerge from the collision unchan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Flaschka
Hermann Flaschka (25 March 1945 – 18 March 2021) was an Austrian-American mathematical physicist and Professor of Mathematics at the University of Arizona, known for his important contributions in completely integrable systems (soliton equations). Childhood Flaschka had lived in the USA since his family immigrated when he was a teenager. They lived in Atlanta, GA. His father Hermenegild Arved Flaschka (1915 - 1999) taught Chemistry at Georgia Tech. Hermann graduated from Druid Hills High School with the class of 1962 and received his Bachelor's degree at Georgia Tech in 1967. Among other achievements there he also received the "William Gilmer Perry Awards for Freshman English" in 1963, despite the fact that he's not a native speaker. Career He received his Ph.D. from the Massachusetts Institute of Technology in 1970. His advisor was Gilbert Strang and the title of his thesis ''Asymptotic Expansions and Hyperbolic Equations with Multiple Characteristics''. He then worked ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax Pair
In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems. Definition A Lax pair is a pair of matrices or operators L(t), P(t) dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation: :\frac= ,L/math> where ,LPL-LP is the commutator. Often, as in the example below, P depends on L in a prescribed way, so this is a nonlinear equation for L as a function of t. Isospectral property It can then be shown that the eigenvalues and more generally the spectrum of ''L'' are independent of ''t''. The matrices/operators ''L'' are said to be '' isospectral'' as t varies. The core observation is that the matrices L(t) are all similar by virtue of :L(t)=U(t,s) L( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Commutator
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted \mathcal_X(T). The differential operator T \mapsto \mathcal_X(T) is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Scattering Transform
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential. The inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems. Overview The inverse scattering transform was first introduced by for the Korteweg–de Vries equation, and soon extended to the nonlinear Schrödinger equation, the Sine-Gordon e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Operator
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi. The name derives from a theorem from Jacobi, dating to 1848, stating that every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix. Self-adjoint Jacobi operators The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers \ell^2(\mathbb). In this case it is given by :Jf_0 = a_0 f_1 + b_0 f_0, \quad Jf_n = a_n f_ + b_n f_n + a_ f_, \quad n>0, where the coefficients are assumed to satisfy :a_n >0, \quad b_n \in \mathbb. The operator will be bounded if and only if the coefficients are bounded. There are close connections with the theory of orthogonal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dispersion (water Waves)
In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium. For a certain water depth, surface gravity waves – i.e. waves occurring at the air–water interface and gravity as the only force restoring it to flatness – propagate faster with increasing wavelength. On the other hand, for a given (fixed) wavelength, gravity waves in deeper water have a larger phase speed than in shallower water. In contrast with the behavior of gravity waves, capillary waves (i.e. only forced by surface tension) propagate faster for shorter wavelengths. Besides frequency dispersion, water waves also exhibit amplitude dispersion. This is a nonlinear effect, by whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bialgebra
In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary. They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group. Lie bialgebras occur naturally in the study of the Yang–Baxter equations. Definition A vector space \mathfrak is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space \mathfrak^* which is compatible. More precisely the Lie algebra structure on \mathfrak is given by a Lie brac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson–Lie Group
In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups. Many quantum groups are quantizations of the Poisson algebra of functions on a Poisson–Lie group. Definition A Poisson–Lie group is a Lie group ''G'' equipped with a Poisson bracket for which the group multiplication \mu:G\times G\to G with \mu(g_1, g_2)=g_1g_2 is a Poisson map, where the manifold G\times G has been given the structure of a product Poisson manifold. Explicitly, the following identity must hold for a Poisson–Lie group: :\ (gg') = \ (g') + \ (g) where f_1 and f_2 are real-valued, smooth functions on the Lie group, while ''g'' and ''g''' are elements of the Lie group. Here, ''L_g'' denotes left-multiplication and ''R_g'' den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
