Electron. J. Diff. Eqns., Vol. 2001(2001), No. 54, pp. 1-18.

Asymptotic behaviour for Schrodinger equations with a quadratic nonlinearity in one-space 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
$$ iu_{t}+\frac{1}{2}u_{xx}=t^{-\alpha}| u_x| ^2,\quad u(0,x) = u_0(x), $$
where $\alpha \in (0,1)$. From the heuristic point of view, solutions to this problem should have a quasilinear character when $\alpha \in (1/2,1)$. We show in this paper that the solutions do not have a quasilinear character for all $\alpha \in (0,1)$. due to the special structure of the nonlinear term. We also prove that for $\alpha \in [1/2,1)$ if the initial data $u_0\in H^{3,0}\cap H^{2,2}$ are small, then the solution has a slow time decay such as $t^{-\alpha /2}$. For $\alpha \in (0,1/2)$, if we assume that the initial data $u_0$ 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 560-0043, Japan
e-mail : nhayashi@math.wani.osaka-u.ac.jp
Pavel I. Naumkin
Instituto de Fisica y Matematicas
Universidad Michoacana, AP 2-82
Morelia, CP 58040, Michoacan, Mexico
e-mail: pavelni@zeus.ccu.umich.mx

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