$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$
In this Chapter we first consider first order PDE and then move to $1$-dimensional wave equation which we analyze by the method of characteristics.
Consider PDE \begin{equation} au_t+bu_x=0. \label{eq-2.1.1} \end{equation} Note that the left-hand expression is a derivative of $u$ along vector field $\ell=(a,b)$. Consider an integral lines of this vector field: \begin{equation} \frac{dt}{a}=\frac{dx}{b}. \label{eq-2.1.2} \end{equation}
Remark 1.
a. Recall from ODE cours that an integral line of the vector field is a line, tangent to it in each point.
b. Often it is called directional derivative but also often then $\ell$ is normalized, replaced by the unit vector of the same direction $\ell^0=\ell/|\ell|$.
If $a$ and $b$ are constant then integral curves are just straight lines $t/a -x/b=C$ where $C$ is a constant along integral curves and it labels them (at least as long as we consider the whole plane $(x,t)$). Therefore $u$ depends only on $C$: \begin{equation} u= \phi \bigl( \frac{t}{a}-\frac{x}{b}\bigr) \label{eq-2.1.3} \end{equation} where $\phi$ is an arbitrary function.
This is a general solution of our equation.
Consider initial value condition $u|_{t=0}=f(x)$. It allows us define $\phi$: $\phi(-x/b)=f(x)\implies \phi (x)= f(-bx)$. Plugging in $u$ we get \begin{equation} u=f\bigl( x-ct\bigr)\qquad\text{with } c=b/a. \label{eq-2.1.4} \end{equation} It is a solution of IVP \begin{equation} \left\{\begin{aligned} &au_t+bu_x=0,\\ &u(x,0)=f(x). \end{aligned} \right. \label{eq-2.1.5} \end{equation} Obviously we need to assume that $a\ne 0$.
If $a=1$ we can rewrite general solution in the form $u(x,t)=\phi_1 (x-bt)$ where $\phi_1(x)=\phi(-x/b)$ is another arbitrary function.
Definition 1. Solutions $u=\chi(x-ct)$ are running waves where $c$ is a propagation speed.
If $a$ and/or $b$ are not constant these integral lines are curves.
Example 1. Consider equation $u_t+tu_x=0$. Then equation of the integral curve is $\frac{dt}{1}=\frac{dx}{t}$ or equivalently $tdt-dx=0$ which solves as $x-\frac{1}{2}t^2=C$ and therefore $u=\phi (x-\frac{1}{2}t^2)$ is a general solution to this equation.
One can see easily that $u=f(x-\frac{1}{2}t^2)$ is a solution of IVP.
Example 2. Consider the same equation but let us consider IVP as $x=0$: $u(0,t)=g(t)$. However it is not a good problem: first, some integral curves intersect line $x=0$ more than once and if in different points of intersection of the same curve initial values are different we get a contradiction (therefore problem is not solvable for $g$ which are not even functions).
On the other hand, if we consider even function $g$ (or equivalently impose initial condition only for $t>0$) then $u$ is not defined on the curves which are not intersecting $x=0$ (which means that $u$ is not defined for $x> \frac{1}{2}t^2$.)
In this example both solvability and unicity are broken.
Consider the same equation albeit with the right-hand expression \begin{equation} au_t+bu_x=f. \label{eq-2.1.6} \end{equation} Then as $\frac{dt}{a}=\frac{dx}{b}$ we have $du = u_t dt + u_xdx = (au_t+bu_x) \frac{dt}{a}=f \frac{dt}{a}$ and therefore we expand our ordinary equation (\ref{eq-2.1.2}) to \begin{equation} \frac{dt}{a}=\frac{dx}{b}=\frac{du}{f}. \label{eq-2.1.7} \end{equation}
Example 3. Consider problem $u_t+u_x=x$. Then $\frac{dx}{1}=\frac{dt}{1}=\frac{du}{x}$. Then $x-t=C$ and $u-\frac{1}{2}x^2=D$ and we get $u-\frac{1}{2}x^2 = \phi (x-t)$ as relation between $C$ and $D$ both of which are constants along integral curves. Here $\phi$ is an arbitrary function. So $u=\frac{1}{2}x^2 + \phi (x-t)$ is a general solution. Imposing Imposing initial condition $u|_{t=0}=0$ (sure, we could impose another condition) we have $\phi(x)=-\frac{1}{2}x^2$ and plugging into $u$ we get $u(x,t)=\frac{1}{2}x^2-\frac{1}{2}(x-t)^2= xt - \frac{1}{2}t^2$.
Example 4. Consider $u_t+ xu_x = x t$. Then $\frac{dt}{1}=\frac{dx}{x}=\frac{du}{xt}$. Solving the first equation $t-\ln x=-\ln C\implies x =Ce^t$ we get integral curves. Now we have \begin{equation*} \frac{du}{xt}=dt \implies du= x t dt= Cte^t dt \implies u=C(t-1)e^t +D = x(t-1)+D \end{equation*} where $D$ must be constant along integral curves and therefore $D=\phi (xe^{-t})$ with an arbitrary function $\phi$. So $u=x(t-1)+\phi (xe^{-t})$ is a general solution of this equation.
Imposing initial condition $u|_{t=0}=0$ (sure, we could impose another condition) we have $\phi(x)=x$ and then $u=x(t-1 +e^{-t})$.
Definition 2. If $a=a(x,t)$ and $b=b(x,t)$ equation is semilinear.
In this case we first define integral curves which do not depend on $u$ and then find $u$ as a solution of ODE along these curves.
Definition 3. Furthermore if $f$ is a linear function of $u$: $f=c(x,t)u + g(x,t)$ original equation is linear.
In this case the last ODE is also linear.
Example 5. Consider $u_t+ xu_x = u$. Then $\frac{dt}{1}=\frac{dx}{x}=\frac{du}{u}$. Solving the first equation $t-\ln x=-\ln C\implies x =Ce^t$ we get integral curves. Now we have \begin{equation*} \frac{du}{u}=dt \implies \ln u= t+\ln D \implies u=De^t=\phi (xe^{-t})e^t \end{equation*} which is a general solution of this equation.
Imposing initial condition $u|_{t=0}=x^2$ (sure, we could impose another condition) we have $\phi(x)= x^2$ and then $u=x^2 e^{-t}$.
Example 6. Consider $u_t+ xu_x = -u^2$. Then $\frac{dt}{1}=\frac{dx}{x}=-\frac{du}{u^2}$. Solving the first equation $x =Ce^t$ we get integral curves. Now we have \begin{equation*} -\frac{du}{u^2}=dt \implies u^{-1}= t+ D \implies u=(t+ \phi(xe^{-t}))^{-1}. \end{equation*} which is a general solution of this equation.
Definition 4. If $a$ and/or $b$ depend on $u$ this is quasininear equation.
For such equations integral curves depend on the solution which can lead to breaking of solution.
Example 7. Consider Burgers equation $u_t+uu_x=0$ (which is an extremely simplified model of gas dynamics. ) We have $\frac{dt}{1}=\frac{dx}{u}=\frac{du}{0}$ and therefore $u=\const$ along integral curves and therefore integral curves are $x-ut=C$.
Consider initial problem $u(x,0)=f(x)$. We take initial point $(y,0)$, find here $u=f(y)$, then $x-f(y)t =y$ (think why?) and we get $u=f(y)$ where $y=y(x,t)$ is a solution of equation $x=f(y)t +y$.
The trouble is that we can define $y$ for all $x$ only if $\frac{\partial }{\partial y}\bigl(f(y)t +y\bigr)$ does not vanish. So, $f'(y)t +1\ne 0$.
This is possible for all $t>0$ if and only if $f'(y)\ge 0$ i.e. $f$ is a monotone non-decreasing function.
So, classical solution breaks if $f$ is not a monotone non-decreasing function. A proper understanding of the global solution for such equation goes well beyond our course.
Example 8. Traffic flow is considered in Appendix
Consider IBVP (initial-boundary value problem) for constant coefficient equation \begin{equation} \left\{\begin{aligned} &u_t +cu_x=0, \qquad &&x>0,\ t>0,\\ &u|_{t=0}= f(x) \qquad &&x>0. \end{aligned}\right. \label{eq-2.1.8} \end{equation}
The general solution is $u=\phi(x-ct)$ and plugging into initial data we get $\phi(x)=f(x)$ (as $x>0$).
So, $u(x,t)= f(x-ct)$. Done!–Not so fast. $f$ is defined only for $x >0$ so $u$ is defined for $x-ct >0$ (or $x >ct$). It covers the whole quadrant if $c\le 0$ (so waves run to the left) and only in this case we are done.
If $c>0$ (waves run to the right) $u$ is not defined as $x< ct$ and to define it here we need a boundary condition at $x=0$. So we get IBVP (initial-boundary value problem) \begin{equation} \left\{\begin{aligned} &u_t +cu_x=0, \qquad &&x>0, t>0,\\ &u|_{t=0}= f(x) \qquad &&x>0,\\ &u|_{x=0}=g(t) \qquad &&t >0. \end{aligned}\right. \label{eq-2.1.9} \end{equation} Then we get $\phi(-ct)=g(t)$ as $t>0$ which implies $\phi(x)=g(-\frac{1}{c}x)$ as $x<0$ and then $u(x,t)=g(-\frac{1}{c}(x-ct))=g(t-\frac{1}{c}x)$ as $x< ct$.
So solution is \begin{equation} u=\left\{\begin{aligned} &f(x-ct)\qquad &&x> c t,\\ &g(t-\frac{1}{c}x)\qquad && x < ct. \end{aligned}\right. \label{eq-2.1.10} \end{equation}
$\Leftarrow$ $\Uparrow$ $\Downarrow$ $\downarrow$ $\Rightarrow$