Differential and Integral Equations

Sharp asymptotics of the small solutions to the nonlinear Schrödinger equations of derivative type

Naoyasu Kita and Takeshi Wada

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper studies the large-time behavior of small solutions to the nonlinear Schrödinger equations in one space dimension. Our relevant equations contain the gauge-invariant cubic nonlinearities of derivative type. Since the nonlinear term is the so-called long-range type, it is well-known that the nonlinear solution tends to the modified linear solution called the first asymptotic term. We present the higher-order asymptotic expansion of the nonlinear solution in weighted $L^2$ and $L^{\infty}$ spaces. The result shows that the nonlinear interaction plays an explicit role in the higher-order asymptotic terms as well as in the phase modification. Our method relies on the nonlinear gauge transformations and the application of $L^{\infty}$ decay estimate by Hayashi--Naumkin [13, 12] for estimating the nonlinear solution in Sobolev spaces.

Article information

Differential Integral Equations, Volume 15, Number 3 (2002), 367-384.

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: 35B40: Asymptotic behavior of solutions 35C20: Asymptotic expansions


Kita, Naoyasu; Wada, Takeshi. Sharp asymptotics of the small solutions to the nonlinear Schrödinger equations of derivative type. Differential Integral Equations 15 (2002), no. 3, 367--384. https://projecteuclid.org/euclid.die/1356060866

Export citation