Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inverse scattering transform for the nonlocal nonlinear Schr\"{o}dinger
  equation with nonzero boundary conditions   In 2013 a new nonlocal symmetry reduction of the well-known AKNS scattering
problem was found; it was shown to give rise to a new nonlocal $PT$ symmetric
and integrable Hamiltonian nonlinear Schr\"{o}dinger (NLS) equation.
Subsequently, the inverse scattering transform was constructed for the case of
rapidly decaying initial data and a family of spatially localized, time
periodic one soliton solution were found. In this paper, the inverse scattering
transform for the nonlocal NLS equation with nonzero boundary conditions at
infinity is presented in the four cases when the data at infinity have constant
amplitudes. The direct and inverse scattering problems are analyzed.
Specifically, the direct problem is formulated, the analytic properties of the
eigenfunctions and scattering data and their symmetries are obtained. The
inverse scattering problem is developed via a left-right Riemann-Hilbert
problem in terms of a suitable uniformization variable and the time dependence
of the scattering data is obtained. This leads to a method to linearize/solve
the Cauchy problem. Pure soliton solutions are discussed and explicit 1-soliton
solution and two 2-soliton solutions are provided for three of the four
different cases corresponding to two different signs of nonlinearity and two
different values of the phase difference between plus and minus infinity. In
the one other case there are no solitons.

15
The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex