Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Classification of Asymptotic Profiles for Nonlinear Schr\"odinger
  Equations with Small Initial Data   We consider a nonlinear Schr\"odinger equation with a bounded local potential
in $R^3$. The linear Hamiltonian is assumed to have two bound states with the
eigenvalues satisfying some resonance condition. Suppose that the initial data
are localized and small in $H^1$. We prove that exactly three local-in-space
behaviors can occur as the time tends to infinity: 1. The solutions vanish; 2.
The solutions converge to nonlinear ground states; 3. The solutions converge to
nonlinear excited states. We also obtain upper bounds for the relaxation in all
three cases. In addition, a matching lower bound for the relaxation to
nonlinear ground states was given for a large set of initial data which is
believed to be generic. Our proof is based on outgoing estimates of the
dispersive waves which measure the relevant time-direction dependent
information of the dispersive wave. These estimates, introduced in [16],
provides the first general notion to measure the out-going tendency of waves in
the setting of nonlinear Schr\"odinger equations.

19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
20
The categories of this paper are: acc-phys