High energy density in multi-soliton collisions

TL;DR

This paper investigates how energy density scales in multi-soliton collisions within the sine-Gordon model, revealing a quadratic relationship with the number of solitons and conditions for maximal energy concentration.

Contribution

It demonstrates that the maximum energy density in multi-soliton collisions scales as the square of the number of solitons and identifies specific conditions for achieving this maximum.

Findings

Maximum energy density scales as N^2 in N-soliton collisions.

Maximal energy density occurs with alternating kinks and antikinks.

Energy type at maximum density depends on whether N is odd or even.

Abstract

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.