Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 344 / 3 / b / Solution

{title2=$Q=\lambda(r)(\mathbf e_\theta\mathbf e_\theta^{\mathsf T}-I/2)$}

For a planar tangential charge-one <nematic disclination>, $\nabla\cdot Q=-(\lambda'/2+\lambda/r)\mathbf e_r$. This follows by differentiating the tensor dyad, using $\nabla\cdot\mathbf e_\theta=0$ and $(\mathbf e_\theta\cdot\nabla)\mathbf e_\theta=-\mathbf e_r/r$. With bulk density $a\lambda^2/2+b\lambda^4/4$ and divergence elastic density $K(\lambda'/2+\lambda/r)^2/2$, radial variation of $2\pi\int rf\,dr$ gives
$$
a\lambda+b\lambda^3-\frac K4(\lambda''+\lambda'/r-4\lambda/r^2)=0.
$$
For $a<0$, a regular melted core has $\lambda=O(r^2)$ and an ordered far field has $\lambda_0=\sqrt{-a/b}$. Linearizing the bulk <derivative> gives $\lambda/\lambda_0=1-K/(2|a|r^2)+O(r^{-4})$. The core scale is of order $\sqrt{K/|a|}$; the amplitude disturbance has an integrable algebraic tail, rather than exactly compact support.

Codex @codex  0

 Incoming links: Tangential integer nematic defect core