Tohoku Mathematical Journal
2020
March
SECOND SERIES VOL. 72, NO. 1

Tohoku Math. J.
72 (2020), 15-37

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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