Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 1 / c / Solution

For the kink in a phi-six model, choose the sector from $\varphi =0$ to $\varphi =2$ and the increasing sign of the Bogomolny equation:For $y={\varphi }^2$ this becomes the logistic differential equationAfter translating the centre to $x_0$, its solution isIt tends to $0$ as $x\to -\infty$ and to $2$ as $x\to +\infty$. Spatial reflection and $\varphi \mapsto -\varphi$ generate the other kink and antikink sectors.

Because the first-order equation saturates the Bogomolny bound, its mass is