# Solving the Sine Gordon PDE in mathematica

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}
]

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## migrated from math.stackexchange.comOct 19 '12 at 18:02

This question came from our site for people studying math at any level and professionals in related fields.

Try to solve an easier PDE first to get a grip on the syntax. – Dr. belisarius Oct 19 '12 at 18:11
your syntax is incorrect; try looking in the docs where they have examples of solving PDEs. – acl Oct 19 '12 at 19:26
This is the Sine Gordon wave equation like you say. Check this out. Also, if you could correct your syntax you might have better luck... – drN Oct 19 '12 at 20:09
I tried editting the equation. However, theres probably insufficient boundary condition and initial condition information to solve this. – drN Oct 19 '12 at 21:37

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]]


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Wow! I love this equation! – drN Oct 19 '12 at 23:05
Vitaliy: Your plot is (as usually) a piece of artwork. Have you written something about creating aesthetically pleasant graphs with Mma? (And if you haven't, could you?) – Dr. belisarius Oct 19 '12 at 23:52
that's a great colour scheme – acl Oct 20 '12 at 0:34
@belisarius thanks, i'm glad you guys like it. i have to find some time - but i'd really love to ;) – Vitaliy Kaurov Oct 20 '12 at 4:44