Internal dynamics of the parametrically excited bound state of double solitary-waves

Abstract

The bound state of double solitary-waves in the parametrically-driven and damped nonlinear Schrödinger system behaves as a classic mechanical oscillator. Here we investigate its internal dynamics including the nonlinear complexity of the solitary-wave interactions. Analytically, it is proved that there is an attractive interaction between two solitary-waves of like polarity (in-phase) and a repulsive interaction between two opposite polarity ones (180°-out-of-phase). Numerical simulation shows that the attracting solitary-waves can form an oscillatory or standing bound state under a proper parametric pumping. The dynamical behaviors of the bound state are then concluded as a stability diagram in the parameter space. It is further revealed that the observed periodic “collisions” of the bound solitonic pulses do not follow the usual collision model in the classic sense, but are actually a dynamical process of the endless dissipative-induced collapses and parametric recreations of the solitary-waves. It is also found that there is a constructive role of damping dissipation in stabilizing and maintaining the regular dynamics of “collisions”. In weakly dissipative media, the parametric excitation of some other internal oscillation of high frequency can have the solitary-wave interaction irregular or even chaotic. Finally, we extend the work and show the existence of the double-pulse bound state in some generalized parametrically driven and damped systems.