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Sinh-Gordon
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and rediscovered by in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions. Origin of the equation and its name There are two equivalent forms of the sine-Gordon equation. In the (real) ''space-time coordinates'', denoted (''x'', ''t''), the equation reads: : \varphi_ - \varphi_ + \sin\varphi = 0, where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (''u'', ''v''), akin to ''asymptotic coordinates'' where : u = \frac, \quad v = \frac, the equation takes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
