The KdV equation on a half-line

The initial boundary value problem on a half-line for the KdV equation with the boundary conditions $u|_{x=0}=a\leq0$, $u_{xx}|_{x=0}=3a^2$ is integrated by means of the inverse scattering method. In order to find the time evolution of the scattering matrix it turned out to be sufficient to solve the Riemann problem on a hyperelliptic curve of genus two, where the conjugation matrices are effectively defined by initial data.