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1], [2], [3], [4], [5], [6], [7],[back]=================================================
Strongly Inelastic Collision of Solitons in Weakly
Perturbed Sine-Gordon Equation
S.V.Dmitriev, K.Abe, T.Shigenari, A.A. Vasiliev(*) and A. E. Miroshnichenko(*)
University of Electro Communications, Japan; Tver State Univiversity, Russia(*)
The sine-Gordon (SG) equation is an exactly integrable nonlinear equation that plays an outstanding role in many applications of physics. The SG equation perturbed by various Hamiltonian and/or dissipative terms has been investigated [1]. Unperturbed SG equation predicts the purely elastic collision between solitons when they recover their initial shapes and there is no energy and momentum exchange. For the SG equation with a nonintegrable term the effect of inelastic many-soliton collision has been found [1]. It is generally supposed that for a small perturbation the degree of inelastisity of collision is small. The aim of this investigation is to show that even for a small perturbation the many-soliton collision can be strongly inelastic [2, 3].
We consider the SG equation as a continuum limit of the Frenkel-Kontorova (F-K) model. The role of perturbation is played by the discreteness of F-K model. The collision between a kink and a high amplitude breather in the F-K chain with a small degree of discreteness was studied numerically and the results were compared with an exact three-soliton solution to the SG equation.
It was found that there exists a narrow range of parameters of quasiparticles where the collision in the discrete system is strongly inelastic. It is conjectured that this effect appears due to the fact that the chain is nonintegrable and a thin chaotic layer exists around separatrix of unperturbed three-soliton solution to the SG equation. In the vicinity of separatrix the influence of the perturbation results in qualitatively new dynamics, namely, in stochastic instability. The width of chaotic layer decreases exponentially as the amplitude of perturbation decreases. It means that in the vicinity of a separatrix there always exists a stochastic region, no matter how small the amplitude of the perturbation is. Physically it means that the many-soliton collision in a weakly perturbed SG system has the probabilistic character.
1. Yu. S. Kivshar and B. A. Malomed: Rev. Mod. Phys. 61, 763 (1989).
2. S. V. Dmitriev, T. Shigenari, A. A. Vasiliev and A. E. Miroshnichenko Phys.Lett. A246 (1998), pp. 129-134.
3. S. V. Dmitriev, T. Shigenari, A. A. Vasiliev and K. Abe: Phys.Rev. B55, 8155 (1997).