Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Existence of full dimensional tori for d-dimensional Nonlinear
  Schr$\ddot{\mbox{O}}$dinger equation   In this paper, we prove the existence of full dimensional tori for
$d$-dimensional nonlinear Schr$\ddot{\mbox{o}}$dinger equation with periodic
boundary conditions \begin{equation*}\label{L1} \sqrt{-1}u_{t}+\Delta
u+V*u\pm\epsilon |u|^2u=0,\hspace{12pt}x\in\mathbb{T}^d,\quad d\geq 1,
\end{equation*} where $V*$ is the convolution potential. Here the radius of the
invariant torus satisfies a slower decay, i.e. \begin{equation*}\label{031601}
I_{\textbf n}\sim e^{-r\ln^{\sigma}\left\|\textbf n\right\|},\qquad \mbox{as}\
\left\|\textbf n\right\|\rightarrow\infty, \end{equation*}for any $\sigma>2$
and $r\geq 1$. This result confirms a conjecture by Bourgain [J. Funct. Anal.
229 (2005), no. 1, 62-94].

0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph