## Differential and Integral Equations

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

#### 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