Let $Q(x, t)$ be an off-diagonal $2 \times 2$ matrix. The matrix NLS equation

$$i Q_{t}-Q_{x x} \sigma_{3}+2 Q^{3} \sigma_{3}=0, \quad \sigma_{3}=\operatorname{diag}(1,-1),$$

admits the Lax pair

$$\begin{aligned} &\mu_{x}+i k\left[\sigma_{3}, \mu\right]=Q \mu \\ &\mu_{t}+2 i k^{2}\left[\sigma_{3}, \mu\right]=\left(2 k Q-i Q^{2} \sigma_{3}-i Q_{x} \sigma_{3}\right) \mu \end{aligned}$$

where $k \in \mathbb{C}, \mu(x, t, k)$ is a $2 \times 2$ matrix and $\left[\sigma_{3}, \mu\right]$ denotes the matrix commutator.

Let $S(k)$ be a $2 \times 2$ matrix-valued function decaying as $|k| \rightarrow \infty$. Let $\mu(x, t, k)$ satisfy the $2 \times 2$-matrix Riemann-Hilbert problem

$$\begin{gathered} \mu^{+}(x, t, k)=\mu^{-}(x, t, k) e^{-i\left(k x+2 k^{2} t\right) \sigma_{3}} S(k) e^{i\left(k x+2 k^{2} t\right) \sigma_{3}}, \quad k \in \mathbb{R} \\ \mu=\operatorname{diag}(1,1)+\mathrm{O}\left(\frac{1}{k}\right), \quad k \rightarrow \infty \end{gathered}$$

(a) Find expressions for $Q(x, t), A(x, t)$ and $B(x, t)$, in terms of the coefficients in the large $k$ expansion of $\mu$, so that $\mu$ solves

$$\mu_{x}+i k\left[\sigma_{3}, \mu\right]-Q \mu=0$$

and

$$\mu_{t}+2 i k^{2}\left[\sigma_{3}, \mu\right]-(k A+B) \mu=0$$

(b) Use the result of (a) to establish that

$$A=2 Q, \quad B=-i\left(Q^{2}+Q_{x}\right) \sigma_{3}$$

(c) Show that the above results provide a linearization of the matrix NLS equation. What is the disadvantage of this approach in comparison with the inverse scattering method?