 Solution

Put $w=\sqrt{Y}$. The separated first integral of a square-root reaction layer gives a primitive $\Phi (Y)$ as defined in solution. One can verify it directly: above the plateau,while below it the derivative has the opposite sign. Accordingly the increasing left profile uses $\Phi (L)=\Phi (0)-\xi$, and the increasing right profile uses $\Phi (R)=\xi$ after choosing its translation constant.

As $w\to k$, the logarithm dominates:Hence the left profile approaches from below with $|\sqrt{L}-k|\propto e^{-\xi /\sqrt{2k}}$, while the right profile approaches from above with $|\sqrt{R}-k|\propto e^{\xi /\sqrt{2k}}$ as $\xi \to -\infty$. These are precisely the required two plateau matching conditions. The absolute value in the logarithm is necessary to use the same real primitive on both sides.