Sine-Gordon equation: Difference between revisions

Content deleted Content added

Inline

Revision as of 11:35, 3 June 2004

The Sine-Gordon equation is a partial differential equation for a function $\phi$ of two real variables, x and t, given as follows:

$\phi _{tt}-\phi _{xx}=\sin \phi ,\,\!$

Another equation is also called the Sine-Gordon equation:

$\phi _{uv}=\sin \phi ,\,\!$

where $\phi$ is again a function of two real varaibles u and v.

The last one is better known in the differential geometry of surfaces. There it is the Gauss integrability of a surface of negative constant Gaussian curvature K given in asymptotic line parameterization (cf. K-surface, pseudospherical surface).

Both PDEs describe solitons.

@@ Line 1: / Line 1: @@
-The '''Sine-Gordon equation''' is a [[partial differential equation]] for a function &phi; of two [[real number|real]] variables, ''x ''and ''t'', given as follows:
+The '''Sine-Gordon equation''' is a [[partial differential equation]] for a function <math>\phi</math> of two [[real number|real]] variables, ''x'' and ''t'', given as follows:
-<math>\frac{d^2\phi}{dt^2}-\frac{d^2\phi}{dx^2}+\sin\phi=0</math>.
+<math>\phi_{tt}- \phi_{xx} = \sin\phi, \,\! </math>
+Another equation is also called the '''Sine-Gordon equation''':
-This [[Partial differential equation|PDE]] describes [[soliton]]s.
+<math>\phi_{uv} = \sin\phi, \,\!</math>
+where <math>\phi</math> is again a function of two real varaibles ''u'' and ''v''.
+The last one is better known in the [[differential geometry]] of surfaces.
+There it is the [[Gauss integrability]] of a surface of negative constant [[Gaussian curvature]] ''K''
+given in asymptotic line parameterization (cf. [[K-surface]], [[pseudospherical surface]]).
+Both [[Partial differential equation|PDEs]] describe [[soliton]]s.
 See also [[B&auml;cklund transform]].