Electron. J. Diff. Eqns., Vol. 2001(2001), No. 54, pp. 118.
Asymptotic behaviour for Schrodinger equations with a quadratic
nonlinearity in onespace dimension
Nakao Hayashi & Pavel I. Naumkin
Abstract:
We consider the Cauchy problem for the Schr\"{o}dinger
equation with a quadratic nonlinearity in one space dimension
where
.
From the heuristic point of view,
solutions to this problem should have a quasilinear character when
.
We show in this paper that the
solutions do not have a quasilinear character for all
.
due to the special structure of the nonlinear term.
We also prove that for
if the initial data
are small, then
the solution has a slow time decay such as
.
For
,
if we assume that the initial data
are
analytic and small, then the same time decay occurs.
Submitted May 22, 2001. Published July 25, 2001.
Math Subject Classifications: 35Q55, 74G10, 74G25.
Key Words: Schrodinger equation, large time behaviour, quadratic nonlinearity.
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Nakao Hayashi
Department of Mathematics
Graduate School of Science
Osaka University
Toyonaka, Osaka 5600043, Japan
email : nhayashi@math.wani.osakau.ac.jp



Pavel I. Naumkin
Instituto de Fisica y Matematicas
Universidad Michoacana, AP 282
Morelia, CP 58040, Michoacan, Mexico
email: pavelni@zeus.ccu.umich.mx 
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