Nonanalytic nonlinear oscillations: Christiaan Huygens, quadratic Schrödinger equations, and solitary waves

Abstract

An example of an oscillatory system with a time-reversible nonanalytic nonlinearity is shown to be a pendulum with a flexible cord sandwiched between two identical circular disks, in contrast to the analytic case of a pendulum interrupted by a single circular disk. The amplitude-dependent frequencies of both cases are perturbatively calculated, and are compared to numerical simulations over the entire range of amplitudes. The nonanalyticity causes the unusual effect of the frequency to vary linearly with amplitude for small amplitudes, which has also been observed in the resonant frequencies of compressional standing waves in sandstone. A general condition for a nonanalytic nonlinearity to yield this behavior is presented. The amplitude-dependent frequency for the double-interrupted pendulum allows an explanation for Huygens'surprising observation that circular interrupters were as effective as cycloidal interrupters in achieving isochronous motion. A lattice of linearly coupled double-interrupted pendulums is described near the lower and upper cutoff modes by quadratic nonlinear Schrödinger ͑NLS͒ equations, in contrast to cubic NLS equations which arise for analytic pendulum lattices as well as typical acoustic and surface waveguides. Solitary breather and kink solutions to the quadratic NLS equations are presented, and are compared to the known soliton solutions of the corresponding cubic NLS equations. Compressional waves in sandstone are shown to be modeled by the inclusion of a nonanalytic quadratic nonlinearity in the stress-strain relationship. Quadratic NLS breathers are predicted to occur in a waveguide of sandstone, and an analysis indicates that such an observation is feasible.