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

Carvajal, Xavier

doi:10.1090/S0033-569X-2012-01297-1

Author: Xavier Carvajal
Journal: Quart. Appl. Math. 71 (2013), 267-281
MSC (2010): Primary 35Q55, 35Q53.
DOI: https://doi.org/10.1090/S0033-569X-2012-01297-1
Published electronically: October 18, 2012
MathSciNet review: 3087422
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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.

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