- #1
Catria
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Homework Statement
Consider a theory with a [itex]\phi^6[/itex]-scalar potential:
[itex]\mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2-\phi^2(\phi^2-1)^2.[/itex]
Why is the solution to the equation of motion not a soliton?
Homework Equations
[itex]\phi''=\frac{\partial V}{\partial\phi}[/itex]
The Attempt at a Solution
[itex]
\phi'\phi''=\phi'\frac{dV}{d\phi}\\
\frac{d}{dx}\left(\frac{\phi'^2}{2}\right)=\frac{dV}{dx}\\
\phi'=\pm \sqrt{2V}\\
\phi'=\phi-\phi^3\\
\Rightarrow\phi(x)= \frac{e^x}{\sqrt{e^{2x}-C_1}}-\frac{e^{-x}}{\sqrt{e^{-2x}-C_2}}[/itex]
Yet the solution on the last line is not a soliton. Why is that so?