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

TL;DR

This paper investigates the ill-posedness of the 5th-order Gardner equation by identifying the Sobolev regularity threshold where solutions become discontinuous with respect to initial data, using new breather solutions.

Contribution

It introduces new breather solutions for the 5th-order Gardner equation and determines the sharp Sobolev index for local well-posedness loss.

Findings

Identified the Sobolev space threshold for ill-posedness.

Demonstrated discontinuity of solution dependence at this threshold.

Provided new insights into the regularity requirements for the equation.

Abstract

We present ill-posedness results for the initial value problem (IVP) 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 $H^s$. 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.