On the ill-posedness of the 5th-order Gardner equation

We present ill-posedness results for the initial value problem of the 5th-order Gardner equation. We use new breather solutions discovered for this higher order Gardner equation to measure the regularity of the Cauchy problem in Sobolev spaces H s ( R ) . We find the sharp Sobolev index under which the local well-posedness of the problem is lost, meaning that the dependence of 5th-order Gardner solutions upon initial data fails to be continuous.