I'm looking at this paper where they work with equations in a moving coordinate system. The following definitions are given
$f(x,t) \equiv f_s(x-ct),\quad g(x,t) \equiv g_s(x-ct)$
As an example, here are the first two terms in the PDE
$f_t + g_x$
where it is understood that the subscript means a partial derivative. That being said, I think they intend for the subscript "s" to mean ordinary differential equation? They do state that the transformation leads to an ODE.
They say, "substituting the definitions into the PDE, integrating once and setting the integration constant equal to zero, we are interested in solutions of the system..."
$-cf_s + g_s$ (this is only the first two terms in the system)
I don't see how they got that expression. Can someone shed some light on the details here. It's been awhile since I've done integration and I'm not sure how to interpret this.
EDIT: Here's a crack at it...
$f_t \equiv \frac{\partial f(x,t)}{\partial t} $
Make the substitution
$\frac{\partial f_s}{\partial t} = \frac{\partial f_s}{\partial X}\frac{\partial X}{\partial t} = \frac{\partial f_s}{\partial X}(-c) $
in which $X = x-ct$. They then integrate with respect to $X$, thus giving
$ \int \frac{\partial f_s}{\partial X}\frac{\partial X}{\partial t} dX = -cf_s$
Same thing with
$\frac{\partial g_s}{\partial x} = \frac{\partial g_s}{\partial X}\frac{\partial X}{\partial x} = \frac{\partial g_s}{\partial X}(1) $
Again, integrate with respect to $X$, thus giving
$ \int \frac{\partial g_s}{\partial X}\frac{\partial X}{\partial x} dX = g_s$
Seems logical?