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2002 Sharp asymptotics of the small solutions to the nonlinear Schrödinger equations of derivative type
Naoyasu Kita, Takeshi Wada
Differential Integral Equations 15(3): 367-384 (2002).

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.

Citation

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Naoyasu Kita. Takeshi Wada. "Sharp asymptotics of the small solutions to the nonlinear Schrödinger equations of derivative type." Differential Integral Equations 15 (3) 367 - 384, 2002.

Information

Published: 2002
First available in Project Euclid: 21 December 2012

zbMATH: 1021.35106
MathSciNet: MR1870648

Subjects:
Primary: 35Q55
Secondary: 35B40 , 35C20

Rights: Copyright © 2002 Khayyam Publishing, Inc.

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Vol.15 • No. 3 • 2002
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