Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 308 / 1

Differentiating the potential givesThus $\varphi =\pm 1$ are stationary points, andbecause $|ϵ|<1$. Both are local minima. Their energies are

For $ϵ=0$, the static energy can be completed to a square:The increasing scalar-field kink therefore obeys the first-order Bogomolny equationWith center $X$, its solution and energy areThe antikink uses the opposite sign.

For small positive $ϵ$, the true vacuum $\varphi =-1$ lies below the false vacuum $\varphi =1$ byThis pressure exerts force $\Delta U$ on a kink with $-1$ on its left and $+1$ on its right. Dividing by its leading mass $4/3$ gives acceleration toward the false-vacuum side:

An antikink followed by a kink encloses a region of the lower vacuum while approaching $\varphi =1$ at both infinities. Vacuum pressure pushes the pair apart, whereas their attraction pulls them together. At a static separation $s$,soThis estimate is self-consistent for $ϵ\ll 1$, when the two soliton cores are well separated.