Reduced Maxwell Bloch Equations Research Articles

The rogue wave type solutions from multiple solitons interactions in the rotating reduced Maxwell–Bloch equations

Rogue waves are usually considered as the large amplitude waves that come from nowhere and disappear without trace. The formation of rogue waves are generally related to the degradation of breather solutions. In some special cases, a special form of the interactions between two solitons can be explained by the breather solutions. The types of soliton collisions are very rich, such as weak interactions, strong interactions, and solitons decay: rogue waves (the large amplitude waves that come from nowhere and disappear without trace), and partial-rogue waves (the large amplitude waves that come from nowhere but leave with a trace). Different from the rogue waves and partial-rogue waves, we mainly study the rogue wave type solutions (the large amplitude waves that come and leave with a trace) from the degeneracy in soliton solutions caused by the interactions of solitons synchronization and resonance in the Rotating reduced Maxwell–Bloch equations, which originated from circularly polarized light propagation on two isotropic electronic field components from optical system. These results systematically illustrate the formation of rogue wave type solutions in terms of boundary conditions, spectral parameters and the number of soliton interactions.

Solitons in a Chiral Medium

The dynamics of a field in a thin waveguide surrounded by helically arranged two-level atoms is studied. The interaction of the induced polarizations of atoms with the field propagating inside the waveguide is described by a system of reduced Maxwell–Bloch equations (RMBEs) in the approximation of unidirectional propagation of the field. Nonlocal dipole–dipole interaction (DDI) of the polarizations of atoms in the helix is described in the approximation of the interaction of nearest neighbors in a curvilinear medium. Soliton solutions of an integrable reduction of the system of equations describing the asymmetric propagation of field pulses in the waveguide in the forward and backward directions are found by the method of Riemann problem with zeros. It is shown that, depending on the chirality sign or the propagation direction, a field pulse in the waveguide can have either the shape of a sharp peak or a nearly rectangular shape. Solutions describing the evolution of field pulses on a nonzero pedestal show that the shape and amplitude of the field pulses can be controlled by the external pumping parameters.

Joint Effect of the Dipole–Dipole Interaction and Constant Dipole Moment on the Shape of the Field Pulse without an Envelope

An integrable model is proposed for a two-level medium with a constant dipole moment, which describes the evolution of electromagnetic field pulses without an envelope with account for the dipole–dipole interaction. In the nearest neighbor approximation, this interaction is taken into account in the form of quadratic dispersion. It is found that such a generalization of the reduced Maxwell–Bloch equations preserves the complete integrability. It is shown using the obtained exact soliton solutions as examples that the joint effect of quadratic dispersion and the constant dipole moment has a number of unique features and provides new opportunities for controlling the shape of field pulses. In particular, it is found that the shape and amplitude of a field pulse depend on the signs of the dipole–dipole interaction as well as on the sign of the constant dipole moment.

Envelope-Free Solitons in a Two-Level Medium Implanted into an Anisotropic Matrix

A model of evolution of field pulses in a medium of two-level atoms implanted into an anisotropic matrix is considered. It is shown that the system of reduced Maxwell–Bloch equations in the approximation of unidirectional propagation can be completely integrable. For a particular case of the model, soliton solutions describing the field pulse evolution are obtained. It is shown that by changing the anisotropy parameters of the external medium, pulse parameters can be controlled.

Integrable generalizations of the sine-Gordon, short pulse, and reduced Maxwell–Bloch equations

The multi-parameter generalizations of the sine-Gordon (SG), short pulse (SP), and reduced Maxwell–Bloch (RMB) equations are considered. These generalizations are integrable by the inverse scattering transformation method and connected with the modified SG equation, its limiting case, and the modified RMB equations by the changes of the dependent and independent variables expressed in terms of the conserved densities and fluxes of the latter. The particular cases of the generalizations considered include the Rabelo–Fokas (RF) equation, the modified SP equation, and some other equations. The properties of the soliton solutions of some of the generalizations are studied. It is revealed that the interaction of the well-defined solitons of the RF equation and the corresponding version of the RMB equations leads to an appearance of the intervals in some regions of the soliton parameters, where the solution becomes multi-valued, and to the blow-up of the solution. In addition, the compacton-like soliton solutions can exist in some cases.

Localized excitations and interactional solutions for the reduced Maxwell-Bloch equations

Based on nonlocal symmetry method, localized excitations and interactional solutions are investigated for the reduced Maxwell–Bloch equations. The nonlocal symmetries of the reduced Maxwell–Bloch equations are obtained by the truncated Painlevé expansion approach and the Möbious invariant property. The nonlocal symmetries are localized to a prolonged system by introducing suitable auxiliary dependent variables. The extended system can be closed and a novel Lie point symmetry system is constructed. By solving the initial value problems, a new type of finite symmetry transformations is obtained to derive periodic waves, Ma breathers and breathers travelling on the background of periodic line waves. Then rich exact interactional solutions are derived between solitary waves and other waves including cnoidal waves, rational waves, Painlevé waves, and periodic waves through similarity reductions. In particular, several new types of localized excitations including rogue waves are found, which stem from the arbitrary function generated in the process of similarity reduction. By computer numerical simulation, the dynamics of these localized excitations and interactional solutions are discussed, which exhibit meaningful structures.

Soliton–cnoidal interactional wave solutions for the reduced Maxwell–Bloch equations**Project supported by the Global Change Research Program of China (Grant No. 2015CB953904), the National Natural Science Foundation of China (Grant Nos. 11675054 and 11435005), the Outstanding Doctoral Dissertation Cultivation Plan of Action (Grant No. YB2016039), and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213). Some

In this paper, we study soliton–cnoidal wave solutions for the reduced Maxwell–Bloch equations. The truncated Painlevé analysis is utilized to generate a consistent Riccati expansion, which leads to solving the reduced Maxwell–Bloch equations with solitary wave, cnoidal periodic wave, and soliton–cnoidal interactional wave solutions in an explicit form. Particularly, the soliton–cnoidal interactional wave solution is obtained for the first time for the reduced Maxwell–Bloch equations. Finally, we present some figures to show properties of the explicit soliton–cnoidal interactional wave solutions as well as some new dynamical phenomena.

Periodic and rational solutions of the reduced Maxwell–Bloch equations

We investigate the reduced Maxwell–Bloch (RMB) equations which describe the propagation of short optical pulses in dielectric materials with resonant non-degenerate transitions. The general Nth-order periodic solutions are provided by means of the Darboux transformation. The Nth-order degenerate periodic and Nth-order rational solutions containing several free parameters with compact determinant representations are derived from two different limiting cases of the obtained general periodic solutions, respectively. Explicit expressions of these solutions from first to second order are presented. Typical nonlinear wave patterns for the four components of the RMB equations such as single-peak, double-peak-double-dip, double-peak and single-dip structures in the second-order rational solutions are shown. This kind of the rational solutions correspond to rogue waves in the reduced Maxwell–Bloch equations.

Reconciling the eigenmode analysis with the Maxwell–Bloch equations approach to superradiance in the linear regime

The superradiance from a slab of inverted two-level atoms is theoretically analyzed in the linear regime from both the perspective of the expansion in eigenfunctions of the integral equation with the Lienard–Wiechert potential as kernel, and that of linearizing the Maxwell–Bloch equations. We show the equivalence of both approaches. We show that the so-called Reduced Maxwell–Bloch equations do not yield even approximately the correct solution when applied in the obvious way, but that they can be made to give the correct solution by adding a fictitious input field.

Reduced Maxwell–Bloch equations with anisotropic dipole momentum

We develop the inverse scattering transform for the recently found integrable system of reduced Maxwell–Bloch equations with two components of polarization and with an anisotropic dipole momentum. The model describes few-cycle pulses of optical or other field propagations. We find that the existence of a nontrivial group of symmetry of the corresponding Lax pair leads to a particular form of the inverse scattering transform equations. We show that solutions can be expressed in terms of the solution of a matrix Riemann–Hilbert problem formulated for the complex plane with a nontrivial group of automorphisms.

An integrable reduction of inhomogeneously broadened optical equations

We introduce a set of reduced Maxwell–Bloch equations that incorporates the effects of a permanent dipole and inhomogeneous broadening. We demonstrate the integrability of this reduction by providing a Lax pair, which is found using the pseudo-potential technique. An appropriate Bäcklund transformation is employed to produce a family of solitonic solutions.

Darboux Transformation and Solitons for Reduced Maxwell–Bloch Equations

The Darboux transformations from arbitrary solutions of reducedMaxwell–Bloch equations is constructed in this article.Hence, in principle, we can find multi-soliton solutions by algebricalgorithm, and the formula will be given explicitly.

Solitons of the reduced Maxwell–Bloch equations for circularly polarized light

It is shown that the reduced Maxwell–Bloch equations for circularly polarized light are integrable by the inverse scattering transform method. A Backlund transformation is given and used to construct a hierarchy of N-soliton solutions. The breather solution corresponding to the 2π-pulse of self-induced transparency is discussed in some detail.

Unidirectional propagation of an ultra-short electromagnetic pulse in a resonant medium with high frequency Stark shift

We consider in the unidirectional approximation the propagation of an ultra short electromagnetic pulse in a resonant medium consisting of molecules characterized by a transition operator with both diagonal and non-diagonal matrix elements. We find the zero-curvature representation of the reduced Maxwell–Bloch equations in the sharp line limit. This can be used to develop the inverse scattering transform method to solve these equations. Finally we obtain two types of exact traveling pulse solutions, one with the usual exponential decay and another with an algebraic decay.

Complete integrability of the reduced Maxwell–Bloch equations with permanent dipole

We obtain the Lax pair, hierarchy of commuting flows and Bäcklund transformations for a reduced Maxwell–Bloch (RMB) system. This system is of particular interest for the description of unipolar, nonoscillating electromagnetic solitons (also called “electromagnetic bubbles” ).

Darboux transformations and coherent interaction of the light pulse with two-level media

Applications of the Darboux transformation method to study the reduced Maxwell-Bloch (RMB) system and of the self-induced transparency (SIT) equations are considered. Both systems describe the propagation of ultrashort optical pulses in a two-level medium to a reasonable approximation. The main result of the present work is the construction of multisoliton solutions on an arbitrary background for RMB and SIT equations. Particular cases of these solutions are discussed in some detail.

The Painleve test, Backlund transformation and solutions of the reduced Maxwell-Bloch equations

The Painleve property for partial differential equations (PDES) proposed by Weiss et al. (1983) is studied for a system of PDES, namely the reduced Maxwell-Bloch (RMB) equations. The RMB equations describe the propagation of short optical pulses through dielectric materials with a resonant non-degenerate transition. The author demonstrates that the RMB system passes the Painleve test, and constructs a Backlund transformation and solutions of the RMB equations.

Rigorous theory of self-induced transparency in the case of a double resonance in a three-level medium

A study is made of steady-state ultrashort electromagnetic radiation pulses of two different wavelengths propagating at the same velocity in a three-level medium under exact resonance conditions. It is shown that such pulses (simultons) correspond to the soliton solutions of the corresponding generalized reduced Maxwell–Bloch equations which thus represent a new example of a system of evolution equations that can be solved exactly by the inverse scattering method. A full characterization of a simulton requires the knowledge of its duration and phase and also of a two-dimensional vector (called the internal polarization vector) with the components equal to the relative amplitudes of the pulses forming a simulton. Collision of two different simultons generally changes their polarizations. An expression is obtained for the angle of rotation of the polarization vectors as a function of the simulton amplitudes and their initial polarizations.

Hamiltonian and recursion operator for the reduced Maxwell-Bloch equations

The Hamiltonian of the reduced Maxwell-Bloch equations (RMBE), an integrable nonlinear evolution equation, is derived. The recursion operator, its inverse and also the hierarchy of higher-order RMBE are obtained.

Theory of self-induced transparency without the approximation of slowly varying amplitudes and phases

An analysis is made of the propagation of an ultrashort optical radiation pulse under conditions of inhomogeneous broadening of a resonance line. By dispensing with the approximation of a slowly varying complex pulse envelope, constraints on the duration are removed and any phase modulation is automatically taken into account. It is shown that if the boundary conditions determining the attenuation of the pulse at its leading edges are defined, the appropriate reduced Maxwell–Bloch equations can be integrated using the method of the inverse problem of scattering theory. Higher conservation laws, whose existence has been doubted, are found.