High energy density in multi-soliton collisions

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.

9 pages, 9 figures

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• Sine-Gordon solitons in networks: Scattering and transmission at vertices
• Classical transitions with the topological number changing in the early Universe
• Scalar triplet on a domain wall: an exact solution
• Kink scattering in hyperbolic models
• False vacuum decay in kink scattering
• Kink-antikink collision in a Lorentz-violating ${\varphi }^4$ model
• Kinks and branes in models with hyperbolic interactions
• A new look at the double sine-Gordon kink-antikink scattering
• Solitary oscillations and multiple antikink-kink pairs in the double sine-Gordon model
• Scattering of kinks in the $B{\phi }^4$ model
• Kink-antikink scattering-induced breathing bound states and oscillons in a parametrized ${\varphi }^4$ model
• Kink scattering in a Lorentz-violating ${\varphi }^6$ model