NONANALYTIC NONLINEAR OSCILLATIONS - HUYGENS,CHRISTIAAN, QUADRATIC SCHRODINGER-EQUATIONS, AND SOLITARY WAVES

An example of an oscillatory system with a time-reversible nonanalytic nonlinearity is shown to be a pendulum with a flexible cord sandwiche d between two identical circular disks, in contrast to the analytic ca se 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 amplit udes. The nonanalyticity causes the unusual effect of the frequency to vary linearly with amplitude for small amplitudes, which has also bee n observed in the resonant frequencies of compressional standing waves in sandstone. A general condition for a nonanalytic nonlinearity to y ield this behavior is presented. The amplitude-dependent frequency for the double-interrupted pendulum allows an explanation for Huygens' su rprising observation that circular interrupters were as effective as c ycloidal interrupters in achieving isochronous motion. A lattice of li nearly coupled double-interrupted pendulums is described near the lowe r and upper cutoff modes by quadratic nonlinear Schrodinger (NLS) equa tions, in contrast to cubic NLS equations which arise for analytic pen dulum lattices as well as typical acoustic and surface waveguides. Sol itary breather and kink solutions to the quadratic NLS equations are p resented, and are compared to the known soliton solutions of the corre sponding cubic NLS equations. Compressional waves in sandstone are sho wn to be modeled by the inclusion of a nonanalytic quadratic nonlinear ity in the stress-strain relationship. Quadratic NLS breathers are pre dicted to occur in a waveguide of sandstone, and an analysis indicates that such an observation is feasible. (C) 1998 Acoustical Society of America. [S0001-4966(98)05309-0].