  In this paper we describe characteristic properties of the scattering data of
the compatible eigenvalue problem for the pair of differential equations
related to the modified Korteweg-de Vries (mKdV) equation whose solution is
defined in some half-strip $(0<x<\infty)\times[0,T]$, or in the quarter plane
$(0<x<\infty)\times(0<t<\infty)$. We suppose that this solution has a
$C^{\infty}$ initial function vanishing as $x\to\infty$, and $C^{\infty}$
boundary values, vanishing as $t\to\infty$ when $T=\infty$. We study the
corresponding scattering problem for the compatible Zakharov-Shabat system of
differential equations associated with the mKdV equation and obtain a
representation of the solution of the mKdV equation through Marchenko integral
equations of the inverse scattering method. The kernel of these equations is
valid only for $x\geq 0$ and it takes into account all specific properties of
the pair of compatible differential equations in the chosen half-strip or in
the quarter plane. The main result is the collection A-B-C of characteristic
properties of the scattering functions given in the paper.
