7 September 1998 Physics Letters A 246 (1998) 1291134 Effect of discreteness on a sine-Gordon three-soliton solution S.V. Dmitriev a,1 , T. Shigenari a , A.A. Vasiliev b , A.E. Miroshnichenko b a Department of Applied Physics and Chemistry, University of Electro-Communications, Chofu-shi, Tokyo 182, Japan b Department of Mathematical Modeling, Tver State University, 33 Zhelyabov Street, 170000 Tver, Russia Received 23 February 1998; revised manuscript received 18 May 1998; accepted for publication 4 June 1998 Communicated by A.R. Bishop Abstract The collision between a kink and a high amplitude breather in the Frenkel1Kontorova chain with a small degree of discreteness was studied numerically and the results were compared with an exact three-soliton solution to the sine-Gordon (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 and that the inelastic collision occurs in the vicinity of a separatrix of the SG three-soliton solution. c  1998 Elsevier Science B.V. Keywords: Sine-Gordon equation; Many-soliton solution; Frenkel1Kontorova chain; Discreteness effect; Stochastic instability; Inelastic collision 1. Introduction In the last few decades the effects of discreteness on the properties of solitary waves have been widely studied [1117]. It has been recognized that the con- tinuum limit approximation cannot describe many ef- fects important in condensed matter physics when the width of the soliton becomes of the order of lattice spacing. The one-soliton solution (kink) has received much attention in most studies. For the static kink in the Frenkel1Kontorova model the exact solution has been obtained [1]. It has been shown that the height of the Peierls1Nabarro (PN) potential decreases rapidly with decreasing the lattice period [2]. The critical pinning velocity and pinning frequency have been de- rived [3] using the perturbation formalism [4]. The 1 Permanent address: General Physics Department, Barnaul State Technical University, 46 Lenin Street, 656099, Barnaul, Russia. change in the shape of a kink and the radiation power loss have been discussed [3]. With the use of the de- rived discretized Hamiltonian formalism the influence of static dressing on the kink pinning frequency and the PN potential have been studied [5,6]. It has been reported that a moving kink not only radiates phonons continuously but can also emit large bursts of phonon radiation [7]. The discreteness strongly affects the Brownian-like motion of the kink and the kink-pair nucleation process in the thermalized chains [8112]. The effects of discreteness on the two-soliton so- lutions have also been studied [12116]. It has been shown that the kink1kink and kink1antikink collisions in a discrete lattice result in an additional radiation of energy [12114]. Using the PN barrier calculations as the base it has been shown that the breather lives extremely long [16]. For the kink1antikink collision in a perturbed SG equation the reflection window ef- fect, first reported for the ϕ 4 -equation [9], has been 0375-9601/98/$ 1 see front matter c  1998 Elsevier Science B.V. All rights reserved. PII S0375-9601 ( 98 ) 00459-9 PLA 8184