  Solitons are very effective in transporting energy over great distances and
collisions between them can produce high energy density spots of relevance to
phase transformations, energy localization and defect formation among others.
It is then important to study how energy density accumulation scales in
multi-soliton collisions. In this study, we demonstrate that the maximal energy
density that can be achieved in collision of $N$ slowly moving kinks and
antikinks in the integrable sine-Gordon field, remarkably, is proportional to
$N^2$, while the total energy of the system is proportional to $N$. This
maximal energy density can be achieved only if the difference between the
number of colliding kinks and antikinks is minimal, i.e., is equal to 0 for
even $N$ and 1 for odd $N$ and if the pattern involves an alternating array of
kinks and anti-kinks. Interestingly, for odd (even) $N$ the maximal energy
density appears in the form of potential (kinetic) energy, while kinetic
(potential) energy is equal to zero. The results of the present study rely on
the analysis of the exact multi-soliton solutions for $N=1,2,$ and 3 and on the
numerical simulation results for $N=4,5,6,$ and 7. Based on these results one
can speculate that the soliton collisions in the sine-Gordon field can, in
principle, controllably produce very high energy density. This can have
important consequences for many physical phenomena described by the
Klein-Gordon equations.
