Electron. J. Diff. Eqns., Vol. 2004(2004), No. 62, pp. 1-16.
### Modified wave operators for nonlinear Schrodinger equations in
one and two dimensions

Nakao Hayashi, Pavel I. Naumkin,
Akihiro Shimomura, & Satoshi Tonegawa

**Abstract:**

We study the asymptotic behavior of solutions, in particular
the scattering theory, for the nonlinear Schrodinger equations
with cubic and quadratic nonlinearities in one or two space
dimensions. The nonlinearities are summation of gauge
invariant term and non-gauge invariant terms. The scattering problem
of these equations belongs to the long range case. We prove the
existence of the modified wave operators to those equations for small
final data. Our result is an improvement of the previous work [13]
Submitted March 10, 2004. Published April 21, 2004.

Math Subject Classifications: 35Q55, 35B40, 35B38

Key Words: Modified wave operators, nonlinear Schrodinger equations

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Nakao Hayashi

Department of Mathematics,
Graduate School of Science

Osaka University, Osaka,
Toyonaka, 560-0043, Japan

email: nhayashi@math.wani.osaka-u.ac.jp
Pavel I. Naumkin

Instituto de Matematicas, UNAM Campus Morelia, AP 61-3 (Xangari)

Morelia CP 58089, Michoacan, Mexico

email: pavelni@matmor.unam.mx

Akihiro Shimomura

Department of Mathematics, Gakushuin University

1-5-1 Mejiro, Toshima-ku, Tokyo 171-8588, Japan

email: simomura@math.gakushuin.ac.jp

Satoshi Tonegawa

College of Science and Technology, Nihon University

1-8-14 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan

email: tonegawa@math.cst.nihon-u.ac.jp

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