On the ill-posedness for a nonlinear Schrödinger-Airy equation

Abstract

Using ideas of Kenig, Ponce and Vega and an explicit solution with two parameters, we prove that the solution map of the initial value problem for a particular nonlinear Schrödinger-Airy equation fails to be uniformly continuous.

Also, we will approximate the solution to the nonlinear Schrödinger-Airy equation by the solution to the cubic nonlinear Schrödinger equation and prove ill-posedness in a more general case than above. This method was originally introduced by Christ, Colliander and Tao for the modified Korteweg-de Vries equation.

Finally, we consider the general case and we prove ill-posedness for all values of the parameters in the equation.