Differential and Integral Equations

Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions

Akihiro Shimomura and Satoshi Tonegawa

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We study scattering theory for nonlinear Schrödinger equations with cubic and quadratic nonlinearities in one and two space dimensions, respectively. For example, the nonlinearities are the sum of the gauge-invariant term and non-gauge-invariant terms such as $\lambda_0 \!|u|^2u +\lambda_1 u^3 +\lambda_2 u\bar{u}^2 +\lambda_3 \bar{u}^3$ in the one-dimensional case, where $\lambda_0 \in {\mathbb R}$ and $\lambda_1,\lambda_2,\lambda_3$ $ \in {\mathbb C}$. The scattering theory for these equations belongs to the long-range case. We show the existence and uniqueness of global solutions for those equations which approach a given modified free profile. The same problem for the nonlinear Schrödinger equation with the Stark potentials is also considered.

Article information

Differential Integral Equations, Volume 17, Number 1-2 (2004), 127-150.

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


Shimomura, Akihiro; Tonegawa, Satoshi. Long-range scattering for nonlinear Schrödinger equations in one and two space dimensions. Differential Integral Equations 17 (2004), no. 1-2, 127--150. https://projecteuclid.org/euclid.die/1356060476

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