Sine-Gordon equation

The sine-Gordon equation is a nonlinear hyperbolic 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 in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space. The equation was 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, and is an example of an integrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the only relativistic system due to its Lorentz invariance. There are two equivalent forms of the sine-Gordon equation. In the (real) space-time coordinates, denoted , the equation reads: where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (u, v), akin to asymptotic coordinates where the equation takes the form This is the original form of the sine-Gordon equation, as it was considered in the 19th century in the course of investigation of surfaces of constant Gaussian curvature K = −1, also called pseudospherical surfaces. There is a distinguished 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 where expresses the angle between the asymptotic lines, and for the second fundamental form, . Then the Gauss–Codazzi equation expressing a compatibility condition between the first and second fundamental forms results in the sine-Gordon equation. This analysis shows that any pseudospherical surface gives rise to a solution of the sine-Gordon equation, although with some caveats: if the surface is complete, it is necessarily singular due to the Hilbert embedding theorem.