Past exam of the theoretical physics major option course of the University of Oxford / 2016 / 5 / a / Solution

Varying the real scalar field gives the Euler-Lagrange field equation ${\partial }_t^2\varphi -{\partial }_x^2\varphi +V^{\prime }(\varphi )=0$. Since $V^{\prime }(\varphi )=a\sin (b\varphi )$, the answer isFor $b\ne 0$, introducing $u=b\varphi$ gives $u_{tt}-u_{xx}+ab\sin u=0$, the Sine-Gordon equation. The proposed real localized kink requires $ab>0$; arbitrary real $a,b$ do not guarantee that condition. In what follows put $k=\sqrt{ab}>0$. The displayed potential itself requires $b\ne 0$, unless a limiting prescription is specified.