The potential is even, so field reflection has $\alpha=-1$; spatial reflection has $\beta=-1$. If $K(x)=\phi^{(0,1)}(x)$, the four oriented interpolating solutions, allowing a common translation, are
$$
\phi^{(0,1)}(x)=K(x),\qquad
\phi^{(1,0)}(x)=K(-x),
$$
$$
\phi^{(0,-1)}(x)=-K(x),\qquad
\phi^{(-1,0)}(x)=-K(-x).
$$
The first and third are kinks under this orientation convention, and spatial reflection gives their antikinks.


Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 313 / 1 / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-313/1/i/solution