Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: High-order numerical method for the nonlinear Helmholtz equation with
  material discontinuities in one space dimension   The nonlinear Helmholtz equation (NLH) models the propagation of
electromagnetic waves in Kerr media, and describes a range of important
phenomena in nonlinear optics and in other areas. In our previous work, we
developed a fourth order method for its numerical solution that involved an
iterative solver based on freezing the nonlinearity. The method enabled a
direct simulation of nonlinear self-focusing in the nonparaxial regime, and a
quantitative prediction of backscattering. However, our simulations showed that
there is a threshold value for the magnitude of the nonlinearity, above which
the iterations diverge. In this study, we numerically solve the one-dimensional
NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity
contains absolute values of the field, the NLH has to be recast as a system of
two real equations in order to apply Newton's method. Our numerical simulations
show that Newton's method converges rapidly and, in contradistinction with the
iterations based on freezing the nonlinearity, enables computations for very
high levels of nonlinearity. In addition, we introduce a novel compact
finite-volume fourth order discretization for the NLH with material
discontinuities.The one-dimensional results of the current paper create a
foundation for the analysis of multi-dimensional problems in the future.

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The categories of this paper are: q-fin
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con