
The sineGordon 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 3space, 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 sineGordon equation. In the (real) ''spacetime coordinates'', denoted (''x'', ''t''), the equation reads: :$\backslash ,\; \backslash varphi\_\; \backslash varphi\_\; +\; \backslash sin\backslash varphi\; =\; 0.$ Passing to the ''light cone coordinates'' (''u'', ''v''), akin to ''asymptotic coordinates'' where : $u=\backslash frac2,\; \backslash quad\; v=\backslash frac2,$ the equation takes the form: :$\backslash varphi\_\; =\; \backslash sin\backslash varphi.\backslash ,$ This is the original form of the sineGordon 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 : $ds^2\; =\; du^2\; +\; 2\backslash cos\backslash varphi\; \backslash ,du\backslash ,\; dv\; +\; dv^2,\backslash ,$ where $\backslash varphi$ expresses the angle between the asymptotic lines, and for the second fundamental form, ''L'' = ''N'' = 0. Then the CodazziMainardi equation expressing a compatibility condition between the first and second fundamental forms results in the sineGordon 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 "sineGordon equation" is a pun on the wellknown Klein–Gordon equation in physics: :$\backslash varphi\_\; \backslash varphi\_\; +\; \backslash varphi\backslash \; =\; 0.\backslash ,$ The sineGordon equation is the Euler–Lagrange equation of the field whose Lagrangian density is given by :$\backslash mathcal\_\backslash text(\backslash varphi)\; =\; \backslash frac(\backslash varphi\_t^2\; \; \backslash varphi\_x^2)\; \; 1\; +\; \backslash cos\backslash varphi.$ Using the Taylor series expansion of the cosine in the Lagrangian, :$\backslash cos(\backslash varphi)\; =\; \backslash sum\_^\backslash infty\; \backslash frac,$ 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 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「SineGordon equation」の詳細全文を読む スポンサード リンク
