Sine-Gordon equation
Hyperbolic partial differential equation / From Wikipedia, the free encyclopedia
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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 Edmond Bour (1862) 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.[1] The equation was rediscovered by Frenkel and Kontorova (1939) in their study of crystal dislocations known as the Frenkel–Kontorova model.[2]
This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions,[3] 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.