High energy density in the collision of $N$ kinks in the $\phi^4$ model

TL;DR

This study demonstrates that in the non-integrable $$ model, the maximal energy density during multi-kink collisions scales quadratically with the number of kinks, similar to the sine-Gordon case, with energy type depending on the parity of the number of kinks.

Contribution

It extends the understanding of energy density scaling in multi-soliton collisions from sine-Gordon to the non-integrable $$ model, including effects of internal modes.

Findings

Maximal energy density scales as N^2 for N kinks.

Odd N collisions predominantly have potential energy density; even N have kinetic.

Internal modes influence the maximal energy density.

Abstract

Recently for the sine-Gordon equation it has been established that during collisions of $N$ slow kinks maximal energy density increases as $N^2$. In this numerical study, the same scaling rule is established for the non-integrable $\phi^4$ model for $N\le 5$. For odd (even) $N$ the maximal energy density is in the form of potential (kinetic) energy density. Maximal elastic strain is also calculated. In addition, the effect of the kink's internal modes on the maximal energy density is analysed for $N=1, 2, 3$. Our results suggest that in multi-soliton collisions very high energy density can be achieved in a controllable manner.