Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stability and instability of breathers in the $U(1)$ Sasa-Satusuma and
  Nonlinear Schr\"odinger models   We consider the Sasa-Satsuma (SS) and Nonlinear Schr\"odinger (NLS) equations
posed along the line, in 1+1 dimensions. Both equations are canonical
integrable $U(1)$ models, with solitons, multi-solitons and breather solutions,
see Yang for instance. For these two equations, we recognize four distinct
localized breather modes: the Sasa-Satsuma for SS, and for NLS the
Satsuma-Yajima, Kuznetsov-Ma and Peregrine breathers. Very little is known
about the stability of these solutions, mainly because of their complex
structure, which does not fit into the classical soliton behavior by
Grillakis-Shatah-Strauss. In this paper we find the natural $H^2$ variational
characterization for each of them, and prove that Sasa-Satsuma breathers are
$H^2$ nonlinearly stable, improving the linear stability property previously
proved by Pelinovsky and Yang. Moreover, in the SS case, we provide an
alternative understanding of the SS solution as a breather, and not only as an
embedded soliton. The method of proof is based in the use of a $H^2$ based
Lyapunov functional, in the spirit of the first and third authors, extended
this time to the vector-valued case. We also provide another rigorous
justification of the instability of the remaining three nonlinear modes
(Satsuma-Yajima, Peregrine y Kuznetsov-Ma), based in the study of their
corresponding linear variational structure (as critical points of a suitable
Lyapunov functional), and complementing the instability results recently proved
e.g. in a paper by the third author.

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The categories of this paper are: hep-ex
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The categories of this paper are: chao-dyn
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The categories of this paper are: chem-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: cs