Find a Lax pair formulation for the linearised NLS equation

$$i q_{t}+q_{x x}=0$$

Use this Lax pair formulation to show that the initial value problem on the infinite line of the linearised NLS equation is associated with the following Riemann-Hilbert problem

$$\begin{array}{ccc} M^{+}(x, t, k) & =M^{-}(x, t, k)\left(\begin{array}{cc} 1 & e^{i k x-i k^{2} t} \hat{q}_{0}(k) \\ 0 & 1 \end{array}\right), & k \in \mathbb{R} \\ M & =\left(\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right)+O\left(\frac{1}{k}\right), & k \rightarrow \infty \end{array}$$

By deforming the above problem obtain the Riemann-Hilbert problem and hence the linear integral equation associated with the following system of nonlinear evolution PDEs

$$\begin{aligned} i q_{t}+q_{x x}-2 \vartheta q^{2} &=0 \\ -i \vartheta_{t}+\vartheta_{x x}-2 \vartheta^{2} q &=0 \end{aligned}$$