Intrinsic localized modes in a three particle Fermi–Pasta–Ulam lattice with on-site harmonic potential

In this paper, a ring of three particle Fermi–Pasta–Ulam lattice with on-site harmonic potential is investigated for the study of intrinsic localized modes (ILMs). In spite of the fact that the integrability of this three particle system has been approved in the absence of cubic interaction and on-site potentials, we show the system remains integrable even in the presence of the on-site harmonic potential. Moreover, we find exact periodic solutions in the form of Jacobi elliptic function, and clarify their correspondences to both stationary and moving ILMs. The dynamics of the system with the inclusion of the cubic term is also explored numerically.