Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The scattering problem for the $abcd$ Boussinesq system in the energy
  space   The Boussinesq $abcd$ system is a 4-parameter set of equations posed in
$\mathbb{R}_t \times \mathbb{R}_x$, originally derived by Bona, Chen and Saut
as first order 2-wave approximations of the incompressible and irrotational,
two dimensional water wave equations in the shallow water wave regime, in the
spirit of the original Boussinesq derivation. Among many particular regimes,
depending each of them in terms of the value of the parameters $(a,b,c,d)$
present in the equations, the "generic" regime is characterized by the setting
$b,d>0$ and $a,c<0$. The system is hamiltonian if also $b=d$. The equations in
this regime are globally well-posed in the energy space $H^1\times H^1$,
provided one works with small solutions. In this paper, we investigate decay
and the scattering problem in this regime, which is characterized as having
(quadratic) long-range nonlinearities, very weak linear decay $O(t^{-1/3})$
because of the one dimensional setting, and existence of non scattering
solutions (solitary waves). We prove, among other results, that for a
sufficiently dispersive $abcd$ systems (characterized only in terms of
parameters $a, b$ and $c$), all small solutions must decay to zero, locally
strongly in the energy space, in proper subset of the light cone $|x|\leq |t|$.
We prove this result by constructing three suitable virial functionals in the
spirit of works by Kowalczyk, Martel and the second author, and more recently
by the last three authors (valid for the simpler scalar "good Boussinesq"
model), leading to global in time decay and control of all local $H^1\times
H^1$ terms. No parity nor extra decay assumptions are needed to prove decay,
only small solutions in the energy space.

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The categories of this paper are: hep-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: cs