System of first order equations, matrix form, quick question

[binbagsss]

Question:

$h_{t}+vh_{x}+v_{x}h=0$
$v_{t}+gh_{x}+vv_{x}=0$

Write it in the form $P_{t}+Q_{x}=0$, where $P=(h,hv)^{T}$,
where $g$ is a constant $>0$, and $v$ and $h$ are functions of $x$ and $t$.

Attempt:

I have $Q=(vh,?)^{T}$, the first equation looks easy enough,

but I'm unsure on the second equation as the component of $P$ for this is giving $hv_{t}+h_{t}v$, whereas I need to get to $v_{t}+gh_{x}+vv_{x}$,

So I'm thinking maybe we need to take a time/x derivaitve of the 2nd equation or multiply it by something. So far multiplication by $h$ seems most promising to me, but I still can't get it, doing this i need the $Q_{x}$ to yield: $h_{t}v+gh_{x}h+vhv_{x}$, I'm pretty sure it's not possible to get the $h_{t}v$ term?

I am correct in thinking you can multiply one of the equations/ take derivatives of one of the equations and not the other right?

Thanks very much !

 

[bigfooted]

Multiply the first equation by v and the second by h and add both equations. Note that $(vh)_t = v_th + vh_t$. Some more hints: how can you rewrite $uu_x =(?)_x$? What is $(uvw)_x$ if you write it out?