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I have derived a solition equation (2 dimensions) from scalar field theory $$\varphi(x) = v\tanh\Bigl(\tfrac{1}{2}m(x - x_0)\Bigr),\tag{1}$$ and also I have got sine-Gordon equation for solition $$\varphi(x) = 4 arc\tanh\Bigl( e^{m\gamma(x-vt) +\delta}\Bigr),\tag{2}$$ Because sine-Gordon equation represents also two dimensions solition then , are these equation represents same? I mean both are sine-Gordon equations?

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What do you mean by "from scalar field theory"? Sine-Gordon is also a scalar field theory. If you mean $\phi^4$ then no - $\phi^4$ and sine-Gordon are different theories and they have different solitons. There is no unique soliton for all two dimensional field theories - each theory has its own spectrum of solitons (none, one or many) just like it has its own spectrum of particle excitations. –  Michael Brown Apr 30 '13 at 6:34
I have got the equation (1) from Mark srednicki, the author got this equation by explaining spontaneous symmetry breaking and then turned into equation (1). Now I'm confused with the sine-Gordon theory which is also 2 dimensional like equation (1). the author also mentioned that, we can find a kink solution from the equation(1). My query is, how can I find a kink solution from equation (1) and can I move the equation (1) to (2)? –  Unlimited Dreamer Apr 30 '13 at 6:42
Can you please write some details the difference between,scalar field theory and $\phi^4$ theory? –  Unlimited Dreamer Apr 30 '13 at 17:30
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$\phi^4$ equation and the sine-Gordon equation are the partial differential equations (PDEs). Equations (1) and (2) are actually the solutions to these PDEs. (BTW, in Eq.(2), it should be trigonometric arctangent, not the hyperbolic one). One can obtain a $\phi^4$ model as an approximation of the sine-Gordon model, expanding the sine term in the equation or cosine term in the Hamiltonian. However, there is no general relation between the $\phi^4$ model and the SG model. In other words, one cannot map any solution of the one equation to that of the other. An absence of the general relation between the two equations follows, for exapmple, from the fact that the SG model is an integrable model, while the $\phi^4$ is not.

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Can you please look the book byLewis H.Rider page no (403). He discussed sine-Gordon equation with explaing the $\phi^4$ terms including which confirms us a relation between $\phi^4$ theory and sine-Gordon equation. –  Unlimited Dreamer Apr 30 '13 at 15:33
Sorry I could not get the book by Rider. But I don't think there is a strict relation between SG and $\phi^4$ models. For example, one can write that $\sin(x) \approx x - x^3/6$, but this does not prove that sinus function is related to a cubic function. BTW, if you take the derivative of your solutions, then eq.(1) gives hyperbolic secant squared, and eq.(2) gives just hyperbolic secant. In this sense, these solutions are related to each other. But in general, the SG model and $\phi^4$ do not have relation in a strict sense. –  user2320292 May 1 '13 at 5:53
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