High energy density in multisoliton 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 sineGordon 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 antikinks. 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 multisoliton solutions for N = 1, 2, and 3 and on the numerical simulation results for N = 4, 5, 6, and 7. The effect of weak Hamiltonian and non-Hamiltonian perturbations on the maximal energy density in multi-kink collisions is also discussed as well as that of the collision relative phase. 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.

Extreme values of elastic strain and energy in sine-Gordon multi-kink collisions

“…This discretization method is fourth order accurate in space [17,32]. Note that in order to increase the accuracy of the integration over the temporal variable the four-step scheme is used with the initial conditions specified for steps j = −3, −2, −1, and 0.…”

“…For instance, in the application of the sG equation to the dislocation theory [46], the interatomic bonds can break if tensile strain is above a threshold value. Thus, the main goal of the present study is to extend the work [32] by the discussion of the extreme values of the two parts of the potential energy density and the extreme values of the tensile and compressive elastic strain in multikink collisions.…”

“…Colliding solitons can produce high energy density spots and for many applications it is important to know how large the energy density can be in multi-soliton collisions. Recently we have addressed this issue in the sG, φ 4 , and φ 6 models [32,33,34]. Basic soliton solutions to sG equation are the kink, which interpolates between nearest wells of the periodic on-site potential, and the breather (or bion), which is a bound state of a kink and antikink [20].…”

“…In the present study we revisit the results presented in [32] aiming to provide more information useful for physical applications. First of all, note that in the Klein-Gordon equations, including sG equation, potential energy consists of two parts, the energy related to the on-site potential and the elastic strain energy.…”

“…First of all, note that in the Klein-Gordon equations, including sG equation, potential energy consists of two parts, the energy related to the on-site potential and the elastic strain energy. In the published work [32] these energies were not separated, but they have very different physical nature and it is important to know how do they contribute to the total potential energy. Secondly, from the physical standpoint it is important to know not only the value of the maximal elastic strain energy density in multi-kink collisions, but also the value of the maximal tensile and compressive strain because the response of media to strain can differ qualitatively depending on the sign of strain.…”