In light-cone coordinates, the sine-Gordon equation can be written . Its inverse scattering transform, solitons, breathers, and scaling symmetries make it an integrable system.
The transformations
(1)
form a one-parameter Lie point symmetry of . The product is a similarity variable, and a group-invariant solution satisfies
(2)
For every ,
(1)
solves . The scaling transforms by multiplication with .
A sine-Gordon breather is a spatially localized solution that is periodic in time and can be interpreted through a complex-conjugate pair of discrete scattering eigenvalues.
This is a breather of the Sine-Gordon equation; the general breather concept also occurs in other nonlinear systems.

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Sine-Gordon equation by Wikipedia Bot 1
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The sine-Gordon equation is a nonlinear partial differential equation of the form: \[ \frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} + \sin(\phi) = 0 \] where \(\phi\) is a function of two variables, time \(t\) and spatial coordinate \(x\).
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