# American Institute of Mathematical Sciences

August  2010, 27(3): 1093-1105. doi: 10.3934/dcds.2010.27.1093

## Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation

 1 Department of Mathematics, Fukuoka University of Education, 1-1 Bunkyoumachi Akama, Munakata City, Fukuoka, 811-4192, Japan

Received  November 2008 Revised  January 2010 Published  March 2010

We consider the fourth order nonlinear Schrödinger type equation (4NLS). The first purpose is to revisit the well-posedness theory of (4NLS). In [8], [9], [20] and [21], they proved the time-local well-posedness of (4NLS) in H *(R) with $s>1/2$ by using the Fourier restriction method. In this paper we give another proof of above result by using simpler approach than the Fourier restriction method. The second purpose is to construct the exact standing wave solution to (4NLS).
Citation: Jun-ichi Segata. Well-posedness and existence of standing waves for the fourth order nonlinear Schrödinger type equation. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1093-1105. doi: 10.3934/dcds.2010.27.1093
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