Ill-posedness for periodic nonlinear dispersive equations

Abstract

In this article, we establish new results about the ill-posedness of the Cauchy problem for the modified Korteweg-de Vries and the defocusing modified Korteweg-de Vries equations, in the periodic case. The lack of local well-posedness is in the sense that the dependence of solutions upon initial data fails to be continuous. We also develop a method for obtaining ill-posedness results in the periodic and non-periodic cases for the equations in the hierarchies of these equations and also in the case of the Benjamin-Ono equation.

Keywords

• Ill-posedness, mKdV equation
• defocusing mKdV equation
• BO equation
• higher order evolutions equations