= Solution
Varying the <real scalar field> gives the <Euler-Lagrange field equation> $\partial_t^2\phi-\partial_x^2\phi+V'(\phi)=0$. Since $V'(\phi)=a\sin(b\phi)$, the answer is
$$
\boxed{\partial_t^2\phi-\partial_x^2\phi+a\sin(b\phi)=0.}
$$
For $b\ne0$, introducing $u=b\phi$ 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 $b\ne0$, unless a limiting prescription is specified.
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