Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local Existence Theory for Derivative Nonlinear Schr\"{o}dinger
  Equations with Non-Integer Power Nonlinearities   We study a derivative nonlinear Schr\"{o}dinger equation, allowing
non-integer powers in the nonlinearity, $|u|^{2\sigma} u_x$. Making careful use
of the energy method, we are able to establish short-time existence of
solutions with initial data in the energy space, $H^1$. For more regular
initial data, we establish not just existence of solutions, but also
well-posedness of the initial value problem. These results hold for real-valued
$\sigma\geq 1,$ while prior existence results in the literature require
integer-valued $\sigma$ or $\sigma$ sufficiently large ($\sigma \geq 5/2$), or
use higher-regularity function spaces.

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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
1
The categories of this paper are: cs
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: eess