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
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. https://doi.org/10.57262/die/1356060866
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