The generalized sine-Gordon equation is obtained under the theoretical investigation of interaction of few-cycle pulses in a nonlinear medium modeled by a set of four-level atoms. This equation is derived without the use of the slowly varying envelope approximation and is shown to be integrable in the frameworks of the inverse scattering transformation method. Its solutions describing the propagation of the solitons and breathers and their interaction are investigated. In the case of different signs of the parameters of the equation considered, it is revealed, in particular, that the collision of solitons with opposite polarities can lead to an appearance of the short-living pulse having extraordinarily large amplitude, whose dynamics is similar to that of rogue waves. Also, the solitons of “rectangular” form and the breathers with rectangular oscillations exist in the case of the same signs of the parameters.

The physical generalization of the system of the reduced Maxwell-Bloch equations describing the nonlinear interaction of the laser pulses with the multilevel quantum medium is obtained without using the slowly varying envelope approximation. It is shown that this system is integrable in the frameworks of the inverse scattering transformation method. The soliton and breather solutions in the form of the unipolar and few-cycle pulses are constructed. Their distinctive features caused solely by the multilevel structure of the medium are discussed. It is revealed that the collision of the solitons can lead to an appearance of the large-amplitude short-living pulse, whose dynamics resembles that of rogue waves.

The propagation of transverse-longitudinal acoustic pulses through a strained cubic crystal containing resonant paramagnetic impurities with effective spin $S=1$ is investigated. It is supposed that the pulses propagate under arbitrary angle with respect to the direction of the external static deformation parallel to the fourth-order symmetry axis. In this geometry, both the transverse and longitudinal components of the acoustic field have high-frequency and zero-frequency spectral components. We show that a pulse can propagate in modes different from the acoustic self-induced transparency. In particular, a pulse propagating in the mode of an acoustic self-induced supertransparency substantially changes the populations of the spin sublevels, but its group velocity remains almost equal to the linear velocity of the sound. If a pulse propagates in the acoustic extraordinary transparency mode, then its group velocity is substantially lower while the sublevel populations remain virtually invariant. Also, the modes of propagation under conditions of weakly excited spin transitions and large detuning of the pulse high-frequency components are identified.

An integrable generalization of the Yajima-Oikawa system of equations has been obtained to describe the optical generation of broadband terahertz radiation in a quadratically nonlinear medium with allowance for the effect of the phase of an optical pulse on this process. The corresponding Lax pair has been found. A soliton solution has been constructed. The analysis has indicated, in particular, that the role of the phase of the optical pulse increases significantly in the vicinity of zero value of the second-order group dispersion parameter. This behavior can result in a significant increase in the efficiency of generation in the soliton regime.

The nonlinear dynamics of a vector two-component optical pulse propagating in quasi-resonance conditions in a medium of nonsymmetric quantum objects is investigated for Stark splitting of quantum energy levels by an external electric field. We consider the case when the ordinary component of the optical pulse induces σ transitions, while the extraordinary component induces the π transition and shifts the frequencies of the allowed transitions due to the dynamic Stark effect. It is found that under Zakharov-Benney resonance conditions, the propagation of the optical pulse is accompanied by generation of an electromagnetic pulse in the terahertz band and is described by the vector generalization of the nonlinear Yajima-Oikawa system. It is shown that this system (as well as its formal generalization with an arbitrary number of optical components) is integrable by the inverse scattering transformation method. The corresponding Darboux transformations are found for obtaining multisoliton solutions. The influence of transverse effects on the propagation of vector solitons is investigated. The conditions under which transverse dynamics leads to self-focusing (defocusing) of solitons are determined.