On the well-posedness of the Eckhaus equation

The Eckhaus equation, which is a nonlinear Schrodinger type equation that can be linearized to the free linear Schrodinger 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 L-2 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. (C) 1997 Published by Elsevier Science B.V.