(a) Consider the group of transformations of $\mathbb{R}^{2}$ given by $g_{1}^{s}:(t, x) \mapsto(\tilde{t}, \tilde{x})=$ $(t, x+s t)$, where $s \in \mathbb{R}$. Show that this acts as a group of Lie symmetries for the equation $d^{2} x / d t^{2}=0$.

(b) Let $\left(\psi_{1}, \psi_{2}\right) \in \mathbb{R}^{2}$ and define $\psi=\psi_{1}+i \psi_{2}$. Show that the vector field $\psi_{1} \partial_{\psi_{2}}-\psi_{2} \partial_{\psi_{1}}$ generates the group of phase rotations $g_{2}^{s}: \psi \rightarrow e^{i s} \psi$.

(c) Show that the transformations of $\mathbb{R}^{2} \times \mathbb{C}$ defined by

$$g^{s}:(t, x, \psi) \mapsto(\tilde{t}, \tilde{x}, \tilde{\psi})=\left(t, x+s t, \psi e^{i s x+i s^{2} t / 2}\right)$$

form a one-parameter group generated by the vector field

$$V=t \partial_{x}+x\left(\psi_{1} \partial_{\psi_{2}}-\psi_{2} \partial_{\psi_{1}}\right)=t \partial_{x}+i x\left(\psi \partial_{\psi}-\psi^{*} \partial_{\psi^{*}}\right)$$

and find the second prolongation $\operatorname{Pr}^{(2)} g^{s}$ of the action of $\left\{g^{s}\right\}$. Hence find the coefficients $\eta^{0}$ and $\eta^{11}$ in the second prolongation of $V$,

$\mathrm{pr}^{(2)} V=t \partial_{x}+\left(i x \psi \partial_{\psi}+\eta^{0} \partial_{\psi_{t}}+\eta^{1} \partial_{\psi_{x}}+\eta^{00} \partial_{\psi_{t t}}+\eta^{01} \partial_{\psi_{x t}}+\eta^{11} \partial_{\psi_{x x}}+\right.$ complex conjugate $)$.

(d) Show that the group $\left\{g^{s}\right\}$ of transformations in part (c) acts as a group of Lie symmetries for the nonlinear Schrödinger equation $i \partial_{t} \psi+\frac{1}{2} \partial_{x}^{2} \psi+|\psi|^{2} \psi=0$. Given that $a e^{i a^{2} t / 2} \operatorname{sech}(a x)$ solves the nonlinear Schrödinger equation for any $a \in \mathbb{R}$, find a solution which describes a solitary wave travelling at arbitrary speed $s \in \mathbb{R}$.