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Solving the soliton equation without energy

In this passage from Srednicki's Quantum Field Theory (page 576) The solution of interest is time independent, so we can set $\dot\varphi = 0$. We can also rewrite the remaining terms in $E$ as ...
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Vortex in D dimensions soliton

let us consider the two-dimensional configuration shown in Fig. 3.1a. The lengths of the arrows represent the magnitude of φ, while their directions indicate the orientation in the $φ_1 -φ_2$ plane. ...