Parametrically excited nonlinear waves and their localizations

The nonlinear wave dynamics in a parametrically resonant and dissipative system is studied both theoretically and numerically. It is found that spatially uniform waves are always subject to saddle-like or modulational instability if the system interval is sufficiently long. The unstable developments lead either to the transitions to solitary-waves or to the parametric amplification of the noidal wave whose wavelength is completely determined by the control strength. The noidal wave itself is shown to be unstable with respect to the breaking of its periodicity. Consequently, a cascade of coalescences occurs among neighboring wave peaks, and finally, the wave becomes highly localized. At a large driving strength, however, there also occurs a reverse process, i.e., the parametric creations of solitonic pulses. As a result, localized pulses are constantly created and annihilated in an irregular way.