Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Two-bubble dynamics for threshold solutions to the wave maps equation   We consider the energy-critical wave maps equation $\mathbb R^{1+2} \to
\mathbb S^2$ in the equivariant case, with equivariance degree $k \geq 2$. It
is known that initial data of energy $ < 8k\pi$ and topological degree zero
leads to global solutions that scatter in both time directions. We consider the
threshold case of energy $8k\pi$. We prove that the solution is defined for all
time and either scatters in both time directions, or converges to a
superposition of two harmonic maps in one time direction and scatters in the
other time direction. In the latter case, we describe the asymptotic behavior
of the scales of the two harmonic maps. The proof combines the classical
concentration-compactness techniques of Kenig-Merle with a modulation analysis
of interactions of two harmonic maps in the absence of excess radiation.

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The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: dg-ga
3
The categories of this paper are: physics
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The categories of this paper are: astro-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: cs
22
The categories of this paper are: chao-dyn