Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the asymptotic stability of the sine-Gordon kink in the energy space   We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a
static, non-symmetric exact solution to SG, stable in the energy space
$H^1\times L^2$. It is well-known that the linearized operator around the kink
has a simple kernel and no internal modes. However, it possesses an odd
resonance at the bottom of the continuum spectrum, deeply related to the
existence of the (in)famous wobbling kink, an explicit periodic-in-time
solution of SG around the kink that contradicts the asymptotic stability of the
kink in the energy space.
  In this paper, we further investigate the influence of resonances in the
asymptotic stability question. We also discuss the relationship between
breathers, wobbling kinks and resonances in the SG setting. By gathering
B\"acklund transformations (BT) and Virial estimates around odd perturbations
of the vacuum solution, we first identify the manifold of initial data around
zero under which BTs are related to the wobbling kink solution. It turns out
that (even) small breathers are deeply related to odd perturbations around the
kink, including the wobbling kink itself. As a consequence of this result,
using BTs we can construct a smooth manifold of initial data close to the kink,
for which there is asymptotic stability in the energy space. The initial data
has spatial symmetry of the form (kink + odd, even), non-resonant in principle,
and not preserved by the flow. This asymptotic stability property holds despite
the existence of wobbling kinks in SG. We also show that wobbling kinks are
orbitally stable under odd data, and clarify some interesting connections
between SG and $\phi^4$ at the level of linear B\"acklund transformations.

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The categories of this paper are: hep-lat
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: q-bio
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The categories of this paper are: quant-ph
0
The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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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: supr-con