Localized Nonlinear Excitations and Interchain Energy Exchange in the Case of Weak Coupling

The problem of energy exchange between weakly coupled nonlinear oscillators is actually far-reaching extension of classical beating problem in linear vibrations theory. Its modern stage goes back to the paper [1] in the field of nonlinear optics (the problem of nonlinear couplers). Then this problem was considered in [2] and [3]. Extension on the case of quasi-harmonic waves in two coupled sin-Gordon chain has been performed in series of papers [4, 5]. A new approach to the problem of non-linear energy exchange was proposed in [6]. In this paper the conception of limiting phase trajectory (LPT), corresponding to complete energy transfer between nonlinear oscillators has been introduced. It was shown that in the framework of this conception an adequate understanding and description of the problem can be obtained with the use of a pair of non-smooth basic functions of time. The proposed approach was also extended on the case of interchain energy exchange by quasi-harmonic waves in weakly coupled oscillatory chains [6]. Recently we presented an efficient use of LPT conception in the case of “small” periodic Fermi-Pasta-Ulam (FPU) chains [7]. Contrary to previous papers, both symmetric and asymmetric interparticle potentials of interaction were considered.

The interchain energy exchange by breathers was first studied both analytically and numerically in [8, 9]. A real possibility of this phenomenon has been shown. However, assumptions were made in that paper which require a justification. In the present paper we reconsidered the problem of interchain energy exchange using the LPT conception and considering subsequently the cases of weakly coupled chains with various degrees of nonlinearity (linear chains, weakly nonlinear chains, chains with nonlinearity compared with coupling).