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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