(i) Write down a Lax pair for the equation

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

Discuss briefly, without giving mathematical details, how this pair can be used to solve the Cauchy problem on the infinite line for this equation. Discuss how this approach can be used to solve the analogous problem for the nonlinear Schrödinger equation.

(ii) Let $q(\zeta, \eta), \tilde{q}(\zeta, \eta)$ satisfy the equations

$$\begin{aligned} &\tilde{q}_{\zeta}=q_{\zeta}+2 \lambda \sin \frac{\tilde{q}+q}{2} \\ &\tilde{q}_{\eta}=-q_{\eta}+\frac{2}{\lambda} \sin \frac{\tilde{q}-q}{2} \end{aligned}$$

where $\lambda$ is a constant.

(a) Show that the above equations are compatible provided that $q, \tilde{q}$ both satisfy the Sine-Gordon equation

$$q_{\zeta \eta}=\sin q$$

(b) Use the above result together with the fact that

$$\int \frac{d x}{\sin x}=\ln \left(\tan \frac{x}{2}\right)+\text { constant }$$

to show that the one-soliton solution of the Sine-Gordon equation is given by

$$\tan \frac{q}{4}=c \exp \left(\lambda \zeta+\frac{\eta}{\lambda}\right)$$

where $c$ is a constant.