 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/iii/paper-344/1/b/solution

For constant

κ

and a planar profile,

μ = f^{'} (ϕ) - κ ϕ^{''} = a ϕ + b ϕ^{3} - κ ϕ^{''} .

Far inside either bulk phase, derivatives vanish and

ϕ = \pm ϕ_{B}

, where

ϕ_{B}^{2} = - a / b

. Therefore

μ = a (\pm ϕ_{B}) + b (\pm ϕ_{B})^{3} = 0.

Since part a makes

μ

constant, it vanishes throughout the interface.

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