= Solution
The Euler–Lagrange equation is $\kappa\phi''=a\phi+b\phi^3$. Multiplication by $\phi'$ and integration gives the first integral
$$
\frac\kappa2(\phi')^2=f(\phi)-f(\phi_B).
$$
With $\phi=\phi_Bg$, $u=x/\xi_0$, and $\xi_0^2=-2\kappa/a$, this becomes
$$
\boxed{2g^2-g^4+(g')^2=1}.
$$
The monotone heteroclinic solutions are
$$
\boxed{
\phi(x)=\pm\phi_B
\tanh\left(\frac{x-x_0}{\xi_0}\right)
}.
$$
<Translation invariance> leaves $x_0$ undetermined in an infinite system. A fixed global composition, boundary condition, or pinning field fixes this midpoint position.
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