Describe the method of characteristics to construct solutions for 1st-order, homogeneous, linear partial differential equations

$$\alpha(x, y) \frac{\partial u}{\partial x}+\beta(x, y) \frac{\partial u}{\partial y}=0$$

with initial data prescribed on a curve $x_{0}(\sigma), y_{0}(\sigma): u\left(x_{0}(\sigma), y_{0}(\sigma)\right)=h(\sigma)$.

Consider the partial differential equation (here the two independent variables are time $t$ and spatial direction $x$ )

$$\frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}=0$$

with initial data $u(t=0, x)=e^{-x^{2}}$.

(i) Calculate the characteristic curves of this equation and show that $u$ remains constant along these curves. Qualitatively sketch the characteristics in the $(x, t)$ diagram, i.e. the $x$ axis is the horizontal and the $t$ axis is the vertical axis.

(ii) Let $\tilde{x}_{0}$ denote the $x$ value of a characteristic at time $t=0$ and thus label the characteristic curves. Let $\tilde{x}$ denote the $x$ value at time $t$ of a characteristic with given $\tilde{x}_{0}$. Show that $\partial \tilde{x} / \partial \tilde{x}_{0}$ becomes a non-monotonic function of $\tilde{x}_{0}$ (at fixed $t$ ) at times $t>\sqrt{e / 2}$, i.e. $\tilde{x}\left(\tilde{x}_{0}\right)$ has a local minimum or maximum. Qualitatively sketch snapshots of the solution $u(t, x)$ for a few fixed values of $t \in[0, \sqrt{e / 2}]$ and briefly interpret the onset of the non-monotonic behaviour of $\tilde{x}\left(\tilde{x}_{0}\right)$ at $t=\sqrt{e / 2}$.