  We prove local existence and uniqueness of solutions of the focusing modified
Korteweg - de Vries equation $u_t + u^2u_x + u_{xxx} = 0$ in classes of
unbounded functions that admit an asymptotic expansion at infinity in
decreasing powers of $x$. We show that an asymptotic solution differs from a
genuine solution by a smooth function that is of Schwartz class with respect to
$x$ and that solves a generalized version of the focusing mKdV equation. The
latter equation is solved by discretization methods.
