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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
KdV Hierarchy
In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which starts with the Korteweg–de Vries equation. Details Let T be translation operator defined on real valued functions as T(g)(x)=g(x+1). Let \mathcal be set of all analytic functions that satisfy T(g)(x)=g(x), i.e. periodic functions of period 1. For each g \in \mathcal, define an operator L_g(\psi)(x) = \psi''(x) + g(x) \psi(x) on the space of smooth functions on \mathbb. We define the Bloch spectrum \mathcal_g to be the set of (\lambda,\alpha) \in \mathbb\times\mathbb^* such that there is a nonzero function \psi with L_g(\psi)=\lambda\psi and T(\psi)=\alpha\psi. The KdV hierarchy is a sequence of nonlinear differential operators D_i: \mathcal \to \mathcal such that for any i we have an analytic function g(x,t) and we define g_t(x) to be g(x,t) and D_i(g_t)= \frac g_t , then \mathcal_g is independent of t. The KdV hierarchy arises naturally as a statement of Huygens' principle for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kadomtsev–Petviashvili Equation
In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili, the KP equation is usually written as: :\displaystyle \partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_u)+\lambda\partial_u=0 where \lambda=\pm 1. The above form shows that the KP equation is a generalization to two spatial dimensions, ''x'' and ''y'', of the one-dimensional Korteweg–de Vries (KdV) equation. To be physically meaningful, the wave propagation direction has to be not-too-far from the ''x'' direction, i.e. with only slow variations of solutions in the ''y'' direction. Like the KdV equation, the KP equation is completely integrable. It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation. In 2002, the regularized version of the KP equation, naturally referre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Davey–Stewartson Equation
In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by to describe the evolution of a three-dimensional wave-packet on water of finite depth. It is a system of partial differential equations for a complex ( wave-amplitude) field u\, and a real ( mean-flow) field \phi\,: :i u_t + c_0 u_ + u_ = c_1 , u, ^2 u + c_2 u \phi_x,\, :\phi_ + c_3 \phi_ = ( , u, ^2 )_x.\, The DSE is an example of a soliton equation in 2+1 dimensions. The corresponding Lax representation for it is given in . In 1+1 dimensions the DSE reduces to the nonlinear Schrödinger equation :i u_t + u_ + 2k , u, ^2 u =0.\, Itself, the DSE is the particular reduction of the Zakharov–Schulman system. On the other hand, the equivalent counterpart of the DSE is the Ishimori equation. The DSE is the result of a multiple-scale analysis of modulated nonlinear surface gravity waves, propagating over a horizontal sea bed. See also * Nonlinear systems * Ishimori equation The Ishimori equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-linear Schrödinger Equation
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benjamin–Ono Equation
In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water. It was introduced by and . The Benjamin–Ono equation is :u_t+uu_x+Hu_=0 where ''H'' is the Hilbert transform. See also * Bretherton equation In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964: :u_+u_+u_+u = u^p, with p integer and p \ge 2. While u_t, u_x and u_ denote partial derivatives of the scalar fiel ... References * * External linksBenjamin-Ono equations: Solitons and Shock Waves Nonlinear partial differential equations Integrable systems {{theoretical-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schrödinger Picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are mostly constant with respect to time (an exception is the Hamiltonian which may change if the potential V changes). This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures. In the Schrödinger picture, the state of a system evolves with time. The evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. For time evolution from a state vector , \psi(t_0)\rangle at time 0 to a state vector , \psi(t)\rangle at time , the time-evolution ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planck Constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalence, the relationship between mass and frequency. Specifically, a photon's energy is equal to its frequency multiplied by the Planck constant. The constant is generally denoted by h. The reduced Planck constant, or Dirac constant, equal to the constant divided by 2 \pi, is denoted by \hbar. In metrology it is used, together with other constants, to define the kilogram, the SI unit of mass. The SI units are defined in such a way that, when the Planck constant is expressed in SI units, it has the exact value The constant was first postulated by Max Planck in 1900 as part of a solution to the ultraviolet catastrophe. At the end of the 19th century, accurate measurements of the spectrum of black body radiation existed, but the distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program, the economic policies advocated by Alexander Hamilton, the first United States Secretary of the Treasury See also * Alexander Hamilton (1755 or 1757–1804), American statesman and one of the Founding Fathers of the US * Hamilton (other) Hamilton may refer to: People * Hamilton (name), a common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question. Quantum mechanics In quantum physics, observables manifest as linear operators on a Hilbert space representing the state space of quantum states. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
