Xtykutl

I'm currently reading about fluid dynamics and the Riemann problem, and a very commonly used equation to introduce the topic is the 1+1D advection equation with constant coefficient $v$:

$$ frac{partial u}{partial t} + v frac{partial u}{partial x} = 0tag{1}$$

for which a solution is
$$ u(x,t) = u(x-vt, 0) = u_0(x-vt) $$
where $u_0 = u(t=0)$ is some initial condition.

This can be easily derived using the method of separation of variables: Let $u(x,t) = f(x)g(y)$.
Then
$$ frac{partial u}{partial t} = f(x) frac{partial g}{partial t}$$

$$ frac{partial u}{partial x} = g(t) frac{partial f}{partial x}
$$
Inserting into the advection equation and restructuring a little, we get

$$frac{1}{g } frac{partial g}{partial t} = frac{1}{f}frac{partial f}{partial x} = -lambda $$

where $lambda$ is some constant. Solving each equation separately gives us

$$ g = K_1 e^{-lambda v t} $$
$$ f = K_2 e^{lambda x} $$
$$ Rightarrow u(x,t) = fg = K e^{lambda (x - vt)} $$

with $K_1$, $K_2$ and $K=K_1 K_2$ are constants stemming from integration.
With
$$u_0 = u(x,t=0) = K e^{lambda x}$$
one can easily see that the solution can be expressed as
$$u(x,t) = u_0(x-vt)$$

So far, so good. Here's my question: Is that the only solution of the 1+1D advection equation with constant coefficients? Is there a proof that this is the only solution?

Yes, it is the only solution. Hints for proof:

1. Go to lightcone coordinates: $x^{pm}~:=~x pm vt$.




2. Show that OP's eq. (1) in 1+1D becomes $frac{partial u}{partial x^+}~=~0$.




3. Deduce that $u=u(x^-)$ is a function of $x^-$ only.

The equation is linear, and the solution to a linear equation in one unknown is always unique.