Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Well-posedness for the fourth-order Schr\"odinger equation with third
  order derivative nonlinearities   We study the Cauchy problem to the semilinear fourth-order Schr\"odinger
equations: \begin{equation}\label{0-1}\tag{4NLS} \begin{cases} i\partial_t
u+\partial_x^4u=G\left(\left\{\partial_x^{k}u\right\}_{k\le
\gamma},\left\{\partial_x^{k}\bar{u}\right\}_{k\le \gamma}\right), & t>0,\ x\in
\mathbb{R}, \\ \ \ \ u|_{t=0}=u_0\in H^s(\mathbb{R}), \end{cases}
\end{equation} where $\gamma\in \{1,2,3\}$ and the unknown function $u=u(t,x)$
is complex valued. In this paper, we consider the nonlinearity $G$ of the
polynomial \[
  G(z)=G(z_1,\cdots,z_{2(\gamma+1)})
  :=\sum_{m\le |\alpha|\le l}C_{\alpha}z^{\alpha}, \] for $z\in
\mathbb{C}^{2(\gamma+1)}$, where $m,l\in\mathbb{N}$ with $3\le m\le l$ and
$C_{\alpha}\in \mathbb{C}$ with $\alpha\in (\mathbb{N}\cup
\{0\})^{2(\gamma+1)}$ is a constant. The purpose of the present paper is to
prove well-posedness of the problem (\ref{0-1}) in the lower order Sobolev
space $H^s(\mathbb{R})$ or with more general nonlinearities than previous
results. Our proof of the main results is based on the contraction mapping
principle on a suitable function space employed by D. Pornnopparath (2018). To
obtain the key linear and bilinear estimates, we construct a suitable
decomposition of the Duhamel term introduced by I. Bejenaru, A. D. Ionescu, C.
E. Kenig, and D. Tataru (2011). Moreover we discuss scattering of global
solutions and the optimality for the regularity of our well-posedness results,
namely we prove that the flow map is not smooth in several cases.

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The categories of this paper are: funct-an
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
18
The categories of this paper are: econ
6
The categories of this paper are: astro-ph
5
The categories of this paper are: gr-qc
22
The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
11
The categories of this paper are: hep-lat