<s>[INST] Finding $u$ for $u_t+uu_x=f$ For a river flowing on a slope, the resistive force is $R = av^2$ and gravitational force, $F = bh$. The flow speed adjusts itself to $av^2=bh \implies v^2=\frac{b}{a}h$. Given the mass conservation law, 
$$h_t + (hv)_x = r.$$
If $v^2=\frac{b}{a}h$,
$$v^2=\frac{b}{a}h\rightarrow (v^2)_t =\frac{b}{a} h_t\rightarrow h_t =\frac{a}{b}(v^2)_t$$
$$(hv)_x=(\frac{a}{b}v^3)_x=\frac{a}{b}3v^2v_x$$
Thus,
$$h_t+(hv)_x=\frac{a}{b}(v^2)_t+\frac{a}{b}3v^2v_x=r$$
$$(v^2)_t+3v^2v_x=r\frac{b}{a}$$
With $u=3\frac{b}{a}h=3v^2$ write,
$$\frac{u_t}{3}+u\frac{(u^{\frac{1}{2}})_x}{3}=r\frac{b}{a}$$
$$\frac{u_t}{3}+u\frac{\frac{1}{2}u_x}{3u^{\frac{1}{2}}}=r\frac{b}{a}$$
$$\frac{u_t}{3}+\frac{u^{\frac{1}{2}}u_x}{6}=r\frac{b}{a}$$
I am trying to get the PDE in the form $u_t+uu_x=f$ (inviscid Burger's equation) but am out of tricks.
 [/INST] $$\frac{u_t}{3}+\frac{u^{\frac{1}{2}}u_x}{6}=r\frac{b}{a} \qquad (1)$$
Supposing $a$ , $b$ and $r$ are constants.
Solving thanks to the method of characteristics :
The system of differential equations for the characteristics curves is :
$$3dt=6\frac{dx}{ u^{\frac{1}{2} }}=\frac{a}{br}du$$
From $\quad 3dt=\frac{a}{br}du\quad$ the equation of a first family of characteristic cuves is :
$$\frac{a}{br}u-3t=c_1$$
From $\quad 6\frac{dx}{u^{\frac{1}{2}}}=\frac{a}{br}du \quad\to\quad 6\frac{br}{a}dx=u^{\frac{1}{2}}du\quad$ the equation of a second family of characteristic cuves is :
$$6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}=c_2$$
The general solution of the PDE $(1)$ is obtained on the form of an implicit equation :
$$\Phi\left(\frac{a}{br}u-3t \; , \; 6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}\right)=0$$ 
where $\Phi$ is any differentiable function of two variables.
Other equivalent forms to express the general solution are :
$$\frac{a}{br}u-3t =F\left( 6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}\right)$$
$$6\frac{br}{a}x-\frac{2}{3}u^{\frac{3}{2}}=G\left( \frac{a}{br}u-3t  \right)$$
where $F$ and $G$ are any differentiable functions, but related ( one is the inverse of the other).
The possibility (or not) to reduce the implicit equation to an explicite equation $u(x,t)$ depends on the kind of function $\Phi$ (or $F$ or $G$).
Unfortunately the boundary conditions are missing in the wording of the question. Without them, one cannot determine the function $\Phi$ (or $F$ or $G$). So, we will not go further.
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