Electron. J. Diff. Equ., Vol. 2012 (2012), No. 162, pp. 1-14.

Nonexistence of asymptotically free solutions to nonlinear Schrodinger systems

Nakao Hayashi, Chunhua Li, Pavel I. Naumkin

Abstract:
We consider the nonlinear Schrodinger systems
$$\displaylines{ -i\partial _tu_1+\frac{1}{2}\Delta u_1=F( u_1,u_2), \cr i\partial _tu_2+\frac{1}{2}\Delta u_2=F( u_1,u_2) }$$
in n space dimensions, where F is a p-th order local or nonlocal nonlinearity smooth up to order p, with $1<p\leq 1+\frac{2}{n}$ for $n\geq 2$ and $1<p\leq 2$ for n=1. These systems are related to higher order nonlinear dispersive wave equations. We prove the non existence of asymptotically free solutions in the critical and sub-critical cases.

Submitted November 17, 2011. Published September 21, 2012.
Math Subject Classifications: 35Q55.
Key Words: Dispersive nonlinear waves; asymptotically free solutions.

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Nakao Hayashi
Department of Mathematics
Graduate School of Science, Osaka University
Osaka, Toyonaka, 560-0043, Japan
email: nhayashi@math.sci.osaka-u.ac.jp
Chunhua Li
Department of Mathematics
Graduate School of Science, Yanbian University
Yanji City, Jilin Province, 133002, China
email: sxlch@ybu.edu.cn
Pavel I. Naumkin
Centro de Ciencias Matemáticas
UNAM Campus Morelia, AP 61-3 (Xangari)
Morelia CP 58089, Michoacán, Mexico
email: pavelni@matmor.unam.mx

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