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

Abstract

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.

Existence regions of longitudinal and transverse intrinsic localized modes in Fermi-Pasta-Ulam chain in two-dimensional plane

“…Each mass can move not only in the axial direction but also in the radial direction. Therefore, the following equations of motion for the masses are derived from Newton's second law of motion [23]:…”

“…The angular frequency of the longitudinal ILM ω L−ILM should be greater than ω x,max . In this case, ω y,max is not related to the existence condition because no motion occurs in the transverse direction [23]. However, for the transverse ILM, both ω x,max and ω y,max should be considered because transverse motion will cause longitudinal motion.…”

“…The difference between lattice dimension and the degree of freedom of each element in the lattice plays a crucial role in ILMs. In fact, novel ILMs have been discovered in FPU chains in three-dimensional space; namely, transverse ILMs and rotating ILMs [23]. In addition, the difference in dimensions makes longitudinal ILMs, which correspond to traditional ILMs in FPU chains, unstable because of buckling effects [23].…”