Solitons with few time or space periods

The theory of solitons in a medium whose dispersion and nonlinearity are described by the Duffing equation, is discussed. A modified Korteweg-de Fries equation is derived for signals comprising a small number of oscillations under the envelope. The soliton and breather propagation is investigated. It is shown that as the number of periods increases, the breather turns into the envelope soliton. The process of formation and propagation of slow solitons in a cubically-nonlinear medium with a limited passband is considered for the case of two real physical systems (the waveguide and periodic structure}. Exact analytical expressions for the soliton shape are obtained and the region of their existence is indicated. The conditions for excitation of a nonpropagating soliton are discussed.