Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long time solutions for quasi-linear Hamiltonian perturbations of
  Schr\"odinger and Klein-Gordon equations on tori   We consider quasi-linear, Hamiltonian perturbations of the cubic
Schr\"odinger and of the cubic (derivative) Klein-Gordon equations on the $d$
dimensional torus. If $\varepsilon\ll1$ is the size of the initial datum, we
prove that the lifespan of solutions is strictly larger than the local
existence time $\varepsilon^{-2}$. More precisely, concerning the Schr\"odinger
equation we show that the lifespan is at least of order $O(\varepsilon^{-4})$,
in the Klein-Gordon case, we prove that the solutions exist at least for a time
of order $O(\varepsilon^{-{8/3}^{-}})$ as soon as $d\geq3$. Regarding the
Klein-Gordon equation, our result presents novelties also in the case of
semi-linear perturbations: we show that the lifespan is at least of order
$O(\varepsilon^{-{10/3}^-})$, improving, for cubic non-linearities and
$d\geq4$, the general results in [17,24].

22
The categories of this paper are: chao-dyn
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
11
The categories of this paper are: hep-lat
29
The categories of this paper are: funct-an