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[tex] \mathcal{L} = 1/2 \partial_{\mu} \varphi \partial^{\mu} \varphi - 1/2 b ( \varphi^{2} - a^{2} )^{2} [/tex], where [tex] a,b \in \mathbb{R}_{>0} [/tex] and [tex] \varphi [/tex] is a real (scalar) field and x are spacetime-coordinates.

I calculated the Euler-Lagrange eq. and get: [tex] \square \varphi + 2 b ( \varphi^{2} - a^{2} ) \varphi = 0 [/tex]

My problem is now to find constant solutions and static ones like [tex] \varphi(x) = f(x-x_{0}) [/tex] where [tex] x_{0} [/tex] is constant. But I don't know how to solve the differential equation above. Does anyone have an idea?