Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dynamics of breathers in the Gardner hierarchy: universality of the
  variational characterization   We present a new variational characterization of breather solutions of any
equation of the \emph{focusing} Gardner hierarchy. This hierarchy is
characterized by a nonnegative index $n$, and $2n+1$ represents the order of
the corresponding PDE member. In this paper, we first show the existence of
such breathers, and that they are solutions of the (2n+1)th-order Gardner
equation. Then we prove a \emph{variational universality property}, in the
sense that all these breather solutions satisfy the \emph{same} fourth order
stationary elliptic ODE, regardless the order of the hierarchy member. This
fact also characterizes them as critical points of the same Lyapunov
functional, that we also construct here. As by product of our approach, we find
breather solutions of the hierarchy of (2n+1)th-order mKdV equations, as well
as a respective characterization of them as solutions of a fourth order
stationary elliptic ODE. We also extend part of these results to the periodic
setting, presenting new breather solutions for the 5th and 7th mKdV members of
the hierarchy. Finally, we prove ill-posedness results for the whole Gardner
hierarchy, by using appropiately their breather solutions.

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The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
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The categories of this paper are: econ
4
The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: solv-int
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat
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The categories of this paper are: bayes-an
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The categories of this paper are: patt-sol
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The categories of this paper are: eess
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The categories of this paper are: acc-phys