Solitary Waves in Nonlinear Models

The chapter begins with a short introduction to solitions and solitary waves in nonlinear dynamics. Thereupon the hierarchical equation and the coupled system are treated separately in the nonlinear case. In both models necessary and sufficient conditions for the existence of solitary waves are established and physical and geometrical properties of the solitary waves are studied. The waves are bell-shaped but asymmetric in the case of presence of the nonlinearity in the micro-scale. In the case of the hierarchical equation, a series axpansion of the wave function is deduced, too. The proofs are based on the canonization of the equations and analysis of trajectories in the phase plane. In some cases nonlinear transforms are applied in order to simplify the treatment of phase curves.