# Domain wall and kink solutions from solitions equations

A general solition equation can be obtaion from scalar field theory $$\varphi(x) = v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{92.6}$$ where $x_0$ is a constant of integration when we drived this solution from Lagrangian.

Comparing to the sine-Gordon theory with this article , can we convert our equation to the sine-Gordon equation?

Because this equation represented for 2 dimensions (1 space dimension and 1 time dimension) then how will I get Kink solutions and domain wall equation from the above equation. Any elaboration would be helpful for me. This is a continuing post of Solving the soliton equation without energy.

ADDITION: If I want to visualize the solition equation (92.6) by graph then what would be the best process?

-
Judging from the previous post you are talking about $\phi^4$ theory. In that case no - you can not convert the solution from one theory to the other as they are entirely different theories. You can however use the same techniques to derive the soliton solution(s) in any 1+1 dimensional scalar field theory. – Michael Brown Apr 30 '13 at 6:38
Second part: why don't you just plot it like any other function? – Michael Brown Apr 30 '13 at 6:39
Ohh got it, which theory is most acceptable? I mean, in future I will deal with higher dimensions but sine-Gordon equation is limited into 2 dimension. So, (92.6) will be better for dimension analysis? – Unlimited Dreamer Apr 30 '13 at 6:46
"which theory is most acceptable?" Neither, or both. Which is to say that it depends on what you are trying to achieve. At a classical level both theories generalise to higher dimensions and the kinks become domain walls. Though sine-Gordon theory is only renormalisable in two dimensions, I believe. $\phi^4$ is renormalisable in four dimensions. – Michael Brown Apr 30 '13 at 6:54
If I want to visualize the solition equation, then how will I do it? – Unlimited Dreamer Apr 30 '13 at 7:18