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The sine-Gordon equation is a second-order nonlinear partial differential equation for a function dependent on two variables typically denoted and , involving the wave operator and the sine of . It was originally introduced by Edmond Bour ( 1862 ) in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation ...
Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. Despite historically it was invented as a single particle equation the Klein–Gordon equation cannot form the basis of a consistent quantum relativistic one-particle theory, any relativistic theory implies creation and ...
Through the superposition principle, given a linear ordinary differential equation (ODE), =, one can first solve =, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
Dirac equation, the relativistic wave equation for electrons and positrons; Gardner equation; Klein–Gordon equation; Knizhnik–Zamolodchikov equations in quantum field theory; Nonlinear Schrödinger equation in quantum mechanics; Schrödinger's equation [2] Schwinger–Dyson equation; Yang-Mills equations in gauge theory
In mathematics, the method of characteristics is a technique for solving partial differential equations.Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.
In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. [1] The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the ...
Multigrid methods can be generalized in many different ways. They can be applied naturally in a time-stepping solution of parabolic partial differential equations, or they can be applied directly to time-dependent partial differential equations. [12] Research on multilevel techniques for hyperbolic partial differential equations is underway. [13]
Large amplitude moving sine-Gordon breather. A breather is a localized periodic solution of either continuous media equations or discrete lattice equations. The exactly solvable sine-Gordon equation [1] and the focusing nonlinear Schrödinger equation [2] are examples of one-dimensional partial differential equations that possess breather ...