On the well-posedness of the Eckhaus equation

The Eckhaus equation, which is a nonlinear Schrödinger type equation that can be linearized to the free linear Schrödinger equation, is considered. A linearized analysis of the nonlinear problem indicates that the periodic boundary value problem is ill-posed. The exact solution also demonstrates the ill-posedness, even though the L² norm of the solution is a constant of the motion. The ill-posedness disappears on the infinite line with square integrable data, but the solitonic solutions, which are not square integrable, are seriously unstable.