Differential and Integral Equations

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

Naoyasu Kita and Takeshi Wada

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Abstract

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

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

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060866

Mathematical Reviews number (MathSciNet)
MR1870648

Zentralblatt MATH identifier
1021.35106

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions 35C20: Asymptotic expansions

Citation

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


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