How to solve the next nonlinear PDE system?

How to solve the next nonlinear PDE system?

Let's have $$ ds^2 = v^2du^2 - dv^2 = dt^2 - dx^2, \quad u = f(x, t),
\quad v = g(x, t). $$ How to get $f, g$? I substituted $u = f(x, t), v =
g(x, t)$ in $ds^2 = u^2dv^2 - du^2$ and then, by equating it to $dt^2 -
dx^2$, got system of PDE: $$ g^2 (\partial_{x}f)(\partial_{t}f) -
(\partial_{x}g) (\partial_{t}g) = 0, \quad g^{2}(\partial_{x}f)^{2} -
(\partial_{x}g)^{2} = -1 , \quad g^{2}(\partial_{t}f)^{2} -
(\partial_{t}g)^{2} = -1 . $$ It may be "simplified" to $$
(\partial_{x}g)^{2} - (\partial_{t}g)^{2} = 1 , \quad
g^{2}(\partial_{t}f)^{2} - (\partial_{x}f)^{2} = 1. $$ How to solve it? Or
maybe there is more simply method to get an expressions for $f, g$?