## Journal of Mathematics of Kyoto University

### Remarks on long range scattering for nonlinear Schrödinger equations with Stark effects

#### Abstract

In this paper, the global existence and asymptotic behavior in time of solutions for the nonlinear Schrödinger equation with the Stark effect in one or two space dimensions are studied. The nonlinearity is cubic and quadratic in one and two dimensional cases, respectively, and it is a summation of a gauge invariant term and non-gauge invariant terms. This nonlinearity is critical between the short range scattering and the long range one. A modified wave operator to this equation is constructed for small final states. Its domain is a certain small ball in $H^{2} \cap \mathcal{F}H^{2}$, where $\mathcal{F}$ is the Fourier transform.

#### Article information

Source
J. Math. Kyoto Univ., Volume 45, Number 1 (2005), 205-216.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250282974

Digital Object Identifier
doi:10.1215/kjm/1250282974

Mathematical Reviews number (MathSciNet)
MR2138807

Zentralblatt MATH identifier
1095.35046

#### Citation

Shimomura, Akihiro; Tonegawa, Satoshi. Remarks on long range scattering for nonlinear Schrödinger equations with Stark effects. J. Math. Kyoto Univ. 45 (2005), no. 1, 205--216. doi:10.1215/kjm/1250282974. https://projecteuclid.org/euclid.kjm/1250282974