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HOME > Table of Contents and Abstracts > Vol. 72, No. 1
Tohoku Mathematical Journal
2020
March
SECOND SERIES VOL. 72, NO. 1
Tohoku Math. J.
72 (2020), 15-37
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Title
HIGHER-ORDER NONLINEAR SCHR\"ODINGER EQUATION IN 2D CASE
Author
Nakao Hayashi and Pavel I. Naumkin
(Received November 8, 2017, revised February 22, 2018) |
Abstract.
We consider the Cauchy problem for the higher-order nonlinear Schrödinger equation in two dimensional case
\[
\left\{\!\!\!
\begin{array}{c}
i\partial _{t}u+\frac{b}{2}\Delta u-\frac{1}{4}\Delta ^{2}u=\lambda
\left\vert u\right\vert u,\text{ }t>0,\text{\ }x\in \mathbb{R}^{2}\,\mathbf{,}
\\
u\left( 0,x\right) =u_{0}\left( x\right) ,\text{\ }x\in \mathbb{R}^{2}
\,\mathbf{,}
\end{array}
\right.
\]
where $\lambda \in \mathbb{R}\mathbf{,}$ $b>0.$ We develop the factorization techniques for studying the large time asymptotics of solutions to the above Cauchy problem. We prove that the asymptotics has a modified character.
Mathematics Subject Classification.
Primary 35Q55; Secondary 35B40.
Key words and phrases.
Higher-order Schrödinger, critical problem, asymptotic behavior, two dimensional.
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