Generalised Inverse Scattering for a Linear PDE Associate to KdV

Inverse Scattering methods for solving integrable nonlinear p.d.e. found their limits as soon as one tried to solve with them new boundary value problems. However, some of these problems, e.g. the quarter-plane problem, can be solved (e.g. by Fokas linear methods), for related linear p.d.e., (e.g. LKdV). It is shown here that a nonlinear algebraic inverse scattering method, which we already applied to nonlinear KdV, but with only partial results, gives the full solution of the quarter-plane problem of another linear p.d.e. associated to KdV. The method makes use of generalised Lax equations and their solutions.