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A theory of an optical breather of self-induced transparency for small area surface plasmon-polariton waves is constructed. The wave equation for an optical nonlinear electric field consisting of surface transverse magnetic modes, traveling along a two-dimensional layer of atomic systems (or semiconductor quantum dots), with a graphene monolayer (or graphene-like two-dimensional material), are shown to reduce to the nonlinear Schr\"odinger equation with damping. It is also shown that damped small intensity surface plasmon-polariton breathers can propagate in such a system and its characteristic parameters depends on the connected media, graphene conductivity, transition layer and transverse structures of the surface plasmon polariton. Explicit analytical expressions for the parameters of an optical surface breather are given. The breather and the soliton in graphene are compared with each other and the differences between their properties are contrasted.

A theory of dispersive soliton of the self-induced transparency in a medium consisting of atoms or semiconductor quantum dots of two types is considered. A two-component medium is modeled by a set of two-level atoms of two types embedded into a conductive host material. These types of atoms correspond to passive atoms (attenuator atoms) and active atoms (amplifier atoms) with inverse population of the energetic levels. The complete solution is given of the Maxwell\char21{}Bloch equations for ensembles of two-type atoms with different parameters and different initial conditions by inverse scattering transform. The solutions of the Maxwell\char21{}Bloch equations for many-component atomic systems by inverse scattering transform are also discussed. The influence of the difference between dipole moments of atoms, the longitudinal and transverse relaxation times, pumping, and conductivity on the soliton is taken into account by means of perturbation theory. The memory effects are described in terms of generalized non-Markovian optical Bloch equations. The condition of a balance between the energy supplied and lost is obtained.

We show that the phenomena of soliton trapping and daughter wave (``shadow'') formation in optical fibers are already contained in the Manakov model [Sov. Phys. JETP 38, 248 (1974)].

The influence of transverse relaxation on the resonance soliton (2 pi hyperbolic-secant pulse) in solids is considered. It is shown that memory effects can be considered in terms of generalized non-Markovian optical Bloch equations. The equations of self-induced transparency with a memory function are presented. Explicit forms for the first-order effects of this relaxation on the frequency shift and the pulse height of a resonant 2 pi soliton in solids are given and discussed. It is shown that memory effects do lead to a qualitative change in the dynamics of the influence of transverse relaxations on a 2 pi pulse in comparison to the McCall-Hahn theory and others. The dynamics of the changes of the inverse width of the pulse are obtained in both the Markovian and the non-Markovian cases with realistic parameters that can be achieved in current experiments. (Less)

Using a singular perturbation expansion, we study the evolution of a Raman loss compensated soliton in an optical fiber. Our analytical results agree quite well with the numerical results of Mollenauer, Gordon, and Islam [IEEE J. Quantum Electron. QE-22, 157 (1986)]. However, there are some differences in that our theory predicts an additional structure that was only partially seen in the numerical calculations. Our analytical results do give a quite good qualitative and quantitative check of the numerical results.

The complete solution of the Maxwell-Bloch equations by the inverse scattering transform as given by Ablowitz, Kaup, and Newell is compared with the McCall-Hahn theory, and others. Not only it is shown that the earlier results are contained in the complete solution, but also several new results are obtained. Among these are an infinite set of "nonlinear moments" which evolve similar to the McCall-Hahn area, a closed-form solution for the nonlinear transmission, how one can determine absolute time delays, why the threshold area for lossless propagation is still exactly $\ensuremath{\pi}$ (even when the initial profile is off resonance but unchirped), and first-order effects of relaxation on $2\ensuremath{\pi}$ hyperbolic-secant pulse propagation. For the last-mentioned results, we find that in general if a $2\ensuremath{\pi}$ hyperbolic-secand pulse is initially off resonance, it will move away from resonance. Also, we find that the McCall-Hahn result for first-order effects on the time delay must be modified for small-amplitude $2\ensuremath{\pi}$ hyperbolic-secant pulses.

It is found that nonlinear coupling of whistler waves with ion-cyclotron fluctuations can lead to spiky whistler electric-field envelopes. Analytical results for the latter, which are solutions with discontinuous derivatives, are given. Possible application of our results in solar-wind plasmas is discussed.