Journal of Mathematics of Kyoto University

Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity

Jun-ichi Segata and Akihiro Shimomura

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Abstract

In this paper, the asymptotic behavior in time of solutions to the one-dimensional fourth order nonlinear Schrödinger type equation with a cubic dissipative nonlinearity $\lambda |u|^{2}u$, where $\lambda$ is a complex constant satisfying $\mathrm{Im}\lambda < 0$, is studied. This nonlinearity is a long-range interaction. The local Cauchy problem at infinite initial time (the final value problem) to this equation is solved for a given final state with no size restriction on it. This implies the existence of a unique solution for the equation approaching some modified free dynamics as $t \to +\infty$ in a suitable function space. Our modified free dynamics decays like $(t\log t)^{-1/2}$ as $t\to \infty$.

Article information

Source
J. Math. Kyoto Univ., Volume 46, Number 2 (2006), 439-456.

Dates
First available in Project Euclid: 14 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1250281786

Digital Object Identifier
doi:10.1215/kjm/1250281786

Mathematical Reviews number (MathSciNet)
MR2284353

Zentralblatt MATH identifier
1116.35113

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35C20: Asymptotic expansions 35P25: Scattering theory [See also 47A40]

Citation

Segata, Jun-ichi; Shimomura, Akihiro. Asymptotics of solutions to the fourth order Schrödinger type equation with a dissipative nonlinearity. J. Math. Kyoto Univ. 46 (2006), no. 2, 439--456. doi:10.1215/kjm/1250281786. https://projecteuclid.org/euclid.kjm/1250281786


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