Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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

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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Additional Information

Xavier Carvajal
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, C.E.P. 21944-970, Rio de Janeiro, R.J. Brazil
Email: carvajal@im.ufrj.br

DOI: https://doi.org/10.1090/S0033-569X-2012-01297-1
Keywords: Schrödinger equation, Korteweg-de Vries equation, ill-posedness
Received by editor(s): June 3, 2011
Published electronically: October 18, 2012
Additional Notes: The author was partially supported by FAPERJ, Brazil under grants E-26/111.564/2008 and E-26/ 110.560/2010 and by the National Council of Technological and Scientific Development (CNPq), Brazil, by the grant 303849/2008-8.
Article copyright: © Copyright 2012 Brown University

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