Characteristic properties of the scattering data for the mKdV equation on the half-line
Abstract
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 is the collection A-B-C of characteristic properties of the scattering functions given in the paper.