Words near each other
 ・ Sinezona wileyi ・ Sinezona zimmeri ・ Sinești ・ Sinești River ・ Sinești River (Bahluieț) ・ Sinești River (Boul) ・ Sinești River (Olteț) ・ Sinești, Ialomița ・ Sinești, Iași ・ Sinești, Ungheni ・ Sinești, Vâlcea ・ Sine Sierra ・ Sine Totoo ・ Sine wave ・ Sine, Arunachal Pradesh ・ Sine-Gordon equation ・ Sine-Saloum ・ Sinea ・ Sinea diadema ・ Sinea spinipes ・ Sinead Burgess ・ Sinead Desmond ・ Sinead Diver ・ Sinead Dudgeon ・ Sinead Farrelly ・ Sinead Harnett ・ Sinead Jennings ・ Sinead Keenan ・ Sinead Kerr ・ Sinead Matthews
 Dictionary Lists
 mini英和辞書
 mini和英辞書
 Webster 1913
 Latin-English
 FOLDOC
 Wikipedia English
 ウィキペディア
 翻訳と辞書　辞書検索 [ 開発暫定版 ]
 スポンサード リンク
 Sine-Gordon equation ： ウィキペディア英語版
Sine-Gordon equation
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. 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:
:$\, \varphi_- \varphi_ + \sin\varphi = 0.$
Passing to the ''light cone coordinates'' (''u'', ''v''), akin to ''asymptotic coordinates'' where
: $u=\frac2, \quad v=\frac2,$
the equation takes the form:
:$\varphi_ = \sin\varphi.\,$
This is the original form of the sine-Gordon equation, as it was considered in the nineteenth century in the course of investigation of surfaces of constant Gaussian curvature ''K'' = −1, also called pseudospherical surfaces. Choose a coordinate system for such a surface in which the coordinate mesh ''u'' = constant, ''v'' = constant is given by the asymptotic lines parameterized with respect to the arc length. The first fundamental form of the surface in these coordinates has a special form
: $ds^2 = du^2 + 2\cos\varphi \,du\, dv + dv^2,\,$
where $\varphi$ expresses the angle between the asymptotic lines, and for the second fundamental form, ''L'' = ''N'' = 0. Then the Codazzi-Mainardi equation expressing a compatibility condition between the first and second fundamental forms results in the sine-Gordon equation. The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations.
The name "sine-Gordon equation" is a pun on the well-known Klein–Gordon equation in physics:
:$\varphi_- \varphi_ + \varphi\ = 0.\,$
The sine-Gordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by
:$\mathcal_\text\left(\varphi\right) = \frac\left(\varphi_t^2 - \varphi_x^2\right) - 1 + \cos\varphi.$
Using the Taylor series expansion of the cosine in the Lagrangian,
:$\cos\left(\varphi\right) = \sum_^\infty \frac,$
it can be rewritten as the Klein–Gordon Lagrangian plus higher order terms
:
\begin
\mathcal_\text(\varphi) & = \frac(\varphi_t^2 - \varphi_x^2) - \frac + \sum_^\infty \frac \\
& = \mathcal_\text(\varphi) + \sum_^\infty \frac.
\end

ウィキペディアで「Sine-Gordon equation」の詳細全文を読む

スポンサード リンク
 翻訳と辞書 : 翻訳のためのインターネットリソース
 Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.