Existence and stability of intrinsic localized modes in a finite FPU chain placed in three-dimensional space

M. Kimura1, A. Mitani1, S. Doi1
1Department of Electrical Engineering, Kyoto University, Katsura, Nishikyo-ku, Kyoto 615-8510, JAPAN
Fermi-Pasta-Ulam (FPU) chain is well known model in which energy localized vibrations exist. Such energy localized vibrations are called intrinsic localized modes (ILMs) or discrete breathers (DBs). This paper discusses existence and stability of ILMs in a finite Fermi-Pasta-Ulam chain which is placed in three-dimensional space, namely, motion of each mass is not constraint to the axis of the chain. First we derive dispersion relations of longitudinal and transversal waves. It is shown that the dispersion relations are changed with respect to the initial stretch or compression of the chain. By using Newton-Raphson method, three kinds of ILM, namely, longitudinal, transversal, and rotating modes, are found in the chain. All masses moves along the axis of the chain in longitudinal modes. The relationship between the frequency and the amplitude distribution completely coincides with ILMs in the traditional FPU chain in which motions of masses are constraint along the axis. On the other hand, main oscillations of transversal modes are perpendicular to the axis. In rotating modes, all masses rotate around the axis. Distribution of the radius of rotations are localized. Stability analysis reveals that almost all the longitudinal and the transversal ILMs are unstable. However, we found a narrow parameter region of the initial stretch in which the transversal ILM becomes stable.
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