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.
Abstract:We model the Fermi-Pasta-Ulam lattice, in which masses move in a twodimensional plane, and identify different types of intrinsic localized modes (ILMs): longitudinal and transverse. The stability of the ILMs is evaluated by using characteristic multipliers. Longitudinal ILMs tend to be unstable because of the buckling effect of the chain. In contrast, transverse ILMs become stable if the chain is initially stretched. This difference between the longitudinal and the transverse ILMs is revealed by computing existence regions with respect to the angular frequency and the initial extension of the chain. The results show that the longitudinal ILMs tend to be stable in low-frequency and low-extension areas whereas the transverse ILMs become stable upon strongly stretching the chain.
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