Abstract
The results of inverse scattering problem associated with the initial-boundary value problem (IBVP) for the Korteweg—de Vries (KdV) equation with dominant surface tension are formulated. The necessary and sufficient conditions for given functions to be the left- and right-reflection coefficients of the scattering problem are established. The time-dependence t, t > 0 of each element of the scattering matrix s(k,t) is found in respective sector of the k-spectral plane by expansion formulas which are constructed from the known initial and boundary conditions of the IBVP. Knowing the right-reflection coefficient calculated from the elements of s(k,t), we solve the Gelfand–Levitan—Marchenko (GLM) equation in the inverse problem. Then the solution of the IBVP is expressible through the solution of the GLM equation. The asymptotic behavior at infinity of time of the solution of the IBVP is shown
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