= Scalar quartic domain wall
{title2=$\phi(x)=M_0\tanh(x/\ell),\quad\ell=\sqrt{2\kappa/|r|}$}
For a <Landau-Ginzburg theory> with positive gradient stiffness $\kappa$, $r<0$ and quartic coupling $u>0$, the degenerate minima are $\pm M_0$, $M_0^2=|r|/u$. A planar <domain wall> solves $\kappa\phi''=r\phi+u\phi^3$. Multiplying by $\phi'$ and using the limiting minima gives $\kappa(\phi')^2/2=u(\phi^2-M_0^2)^2/4$. Separating this equation yields the displayed profile, and integrating its excess energy gives the positive surface tension $2\sqrt{2\kappa}|r|^{3/2}/(3u)$. Its area cost is subextensive compared with bulk volume, permitting phase-separated <common-tangent construction> states when the average <order parameter> is constrained.
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