Characteristic properties of the scattering data for the mKdV equation on the half-line

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 < ∞) × [0, T ], or in the quarter plane (0 < x < ∞) × (0 < t < ∞). We suppose that this solution has a C∞ initial function vanishing as x → ∞, and C∞ boundary values, vanishing as t → ∞ when T = ∞. 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 ≥ 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 of the paper is the collection A–B–C of characteristic properties of the scattering functions given below.