Solving the Sine Gordon PDE in mathematica

Question

how can i solve this equation in mathematica? this is sine-gordon eq. but the boundary condition can not recognized by mathematica . thank you for you attention.

NDSolve[{∂_(x,x) F[t,x]==(1-ζ)(Sin[F[t,x]]+β∂_t F[t,x]+∂_(t,t) F[t,x]),...
(∂_x F[t,x]==∂_t  F[t,x]/.x→0),(∂_x F[t,x]==∂_t F[t,x]/.x→1)},F,{t,0,10},{x,0,1}]

Edit:

\[Beta] = 1;
\[Zeta] = 1;
NDSolve[
 {
  D[f[t, x], x, 
    x] == (1 - \[Zeta]) Sin[[t, x]] + \[Beta] D[f[t, x], t] + 
    D[f[t, x], t, t],
  D[f[t, 0], x] == D[f[t, 0], t],
  D[f[t, 1], x] == D[f[t, 1], t]

  },
 f,
 {t, 0, 10},
 {x, 0, 1}
 ]

Answer 1

As the comments say, you really have to learn correct syntax.

Set up Sine-Gordon equation with initial and boundary conditions:

eq = {D[u[t, x], {t, 2}] == Sin[u[t, x]] + D[u[t, x], {x, 2}], 
  u[0, x] == E^(-x^2), Derivative[1, 0][u][0, x] == 0, 
  u[t, -10] == u[t, 10]};

Solve it:

sol = NDSolve[eq, u, {t, 0, 30}, {x, -10, 10}];

Plot it:

Plot3D[Evaluate[u[t, x] /. sol], {t, 0, 20}, {x, -10, 10}, 
 PlotRange -> All, PlotStyle -> Directive[Specularity[White, 20], Opacity[0.8]], 
 ColorFunction -> "DarkRainbow", PlotPoints -> 50, MeshStyle -> Opacity[.5]]