This work is devoted to systematically study general $N$-soliton solutions possibly containing multiple degenerate soliton groups (DSGs), in the context of the sharp-line Maxwell-Bloch equations with a zero background.We also show that results can be readily migrated to other integrable systems, with the same non-self-adjoint Zakharov-Shabat scattering problem or alike. Results for the focusing nonlinear Schrödinger equation and the complex modified Korteweg-De Vries equation are obtained as explicit examples for demonstrative purposes. A DSG is a localized coherent nonlinear traveling-wave structure, comprised of inseparable solitons with identical velocities. Hence, DSGs are generalizations of single solitons (considered as $1$-DSGs), and form fundamental building blocks of solutions of many integrable systems. We provide an explicit formula for an $N$-DSG and its center. With the help of the Deift-Zhou's nonlinear steepest descent method, we prove the localization of DSGs, and calculate the long-time asymptotics for an arbitrary $N$-soliton solutions. It is shown that the solution becomes a linear combination of multiple DSGs in the distant past and future, with explicit formulae for the asymptotic phase shift for each DSG. Other generalizations of a single soliton are also discussed, such as $N$th-order solitons and soliton gases. We prove that every $N$th-order soliton can be obtained by fusion of eigenvalues of $N$-soliton solutions, with proper rescalings of norming constants, and demonstrate that soliton-gas solution can be considered as limits of $N$-soliton solutions as $N\to+\infty$.
A new Hamiltonian structure of the Maxwell-Bloch equations is described. In this setting the Maxwell-Bloch equations appear as a member of a family of generalized Maxwell-Bloch systems. The family is parameterized by compact semi-simple Lie groups, the original Maxwell-Bloch system being the member corresponding to SU(2). The Hamiltonian structure is then used in the construction of a new family of symmetries and the associated conserved quantities of the Maxwell-Bloch equations.
The (2+1)-dimensional Hirota-Maxwell-Bloch equation (HMBE) is integrable by the Inverse Scattering Method. In this paper, we construct a Darboux transformation (DT) of the (2+1)-dimensional HMBE. Also the one-soliton solution obtained by means of the one-fold DT. For the $n$-soliton solution the general form is presented.
In this paper, we construct a Darboux transformation (DT) of the (2+1)-dimensional Schrödinger-Maxwell-Bloch equation (SMBE) which is integrable by the Inverse Scattering Method. Using this DT, the one-soliton solution and periodic solution are obtained from the "seed" solutions.
In this paper, we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB) equations which is governed by femtosecond pulse propagation through inhomogeneous doped fibre. The determinant representation of Darboux transformation is used to derive soliton solutions, positon solutions of the IH-MB equations.
In this paper, we derive n-fold Darboux transformation of the two-component Hirota and the Maxwell-Bloch(TH-MB) equations and its determinant representation. Using Darboux determinant representation, we provide soliton solutions, positon solutions of the TH-MB equations.
We report for the first time exact solutions of a completely integrable nonlinear lattice system for which the dynamical variables satisfy a q-deformed Lie algebra - the Lie-Poisson algebra su_q(2). The system considered is a q-deformed lattice for which in continuum limit the equations of motion become the envelope Maxwell-Bloch (or SIT) equations describing the resonant interaction of light with a nonlinear dielectric. Thus the N-soliton solutions we here report are the natural q-deformations, necessary for a lattice, of the well-known multi-soliton and breather solutions of self-induced tra…
In this paper, based on the Hirota and Maxwell-Bloch (H-MB) system and its application in the theory of the femtosecond pulse propagation through an erbium doped fiber, we define two kinds of nonlocal Hirota and Maxwell-Bloch (NH-MB) systems, namely, $PT$-symmetric NH-MB system and reverse space-time NH-MB system. Then we construct the Darboux transformations of these NH-MB systems. Meanwhile, we derive the explicit solutions by the Darboux transformations.
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, and from two different limiting cases of the obtained general periodic solutions, the Nth-order degenerate periodic and Nth-order rational solutions containing several free parameters with compact determinant representations are derived, respectively. Explicit expressions of these solutions from first to second order are prese…
The Baker-Akhiezer (BA) function theory was successfully developed in the mid 1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and theory of completely integrable nonlinear equations such as Korteweg-de Vries equation, nonlinear Schrödinger equation, sine-Gordon equation, Kadomtsev-Petviashvili equation. Subsequently the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch (MB) system or for the Karpman-Kaup equations, which conta…
We consider the quantum nonlinear Schrödinger equation in one space and one time dimension. We are interested in the non-free-fermionic case. We consider static temperature-dependent correlation functions. The determinant representation for correlation functions simplifies in the small mass limit of the Bose particle. In this limit we describe the correlation functions by the vacuum expectation value of a boson-valued solution for Maxwell-Bloch differential equation. We evaluate long-distance asymptotics of correlation functions in the small mass limit.
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 Painleve expansion approach and the Mobious 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 …
In this paper the coupled Maxwell-Bloch equations which describe the propagation of two optical pulses in an optical medium with coherent three-level atoms are studied by Darboux transformation. The general nth-order rogue wave solution involving two different choices of multiple roots for the spectral characteristic equation and the multiparametric nth-order semirational solution are both obtained in terms of Schur polynomials. The explicit rogue wave solutions and semirational solutions from first to second order are provided. In contrast to the known Peregrine soliton, dark and four-petaled…
In this paper, we derive a Darboux transformation of the Hirota and the Maxwell-Bloch(H-MB) system which is governed by femtosecond pulse propagation through an erbium doped fibre and further generalize it to the matrix form of the $n$-fold Darboux transformation of this system. This $n$-fold Darboux transformation implies the determinant representation of $n$-th new solutions of $(E^{[n]},p^{[n]}, η^{[n]})$ generated from known solution of $(E, p,η)$. The determinant representation of $(E^{[n]},p^{[n]} ,η^{[n]})$ provides soliton solutions, positon solutions, and breather solutions (both brig…
We derive and study the hyperbolic reduced Maxwell-Bloch equations (HRMB), a simplified model for the dynamics of strongly correlated Bose-Einstein condensates (BECs), and in particular for the interaction between the BEC atoms and its evaporated atoms under the strong interactions. This equation is one among four which are proven to be integrable via the existence of a Lax pair, and thus the method of inverse scattering transform. Another equation is the reduced Maxwell-Bloch equation of quantum optics and the two others do not have physical applications yet. By studying the linear stability …
We study the (characteristic) Cauchy problem for the Maxwell-Bloch equations of light-matter interaction via asymptotics, under assumptions that prevent the generation of solitons. Our analysis clarifies some features of the sense in which physically-motivated initial/boundary conditions are satisfied. In particular, we present a proper Riemann-Hilbert problem that generates the unique causal solution to the Cauchy problem, that is, the solution vanishes outside of the light cone. Inside the light cone, we relate the leading-order asymptotics to self-similar solutions that satisfy a system of …
We find the Lax pair for a system of reduced Maxwell-Bloch equations that describes the propagation of two-component extremely short electromagnetic pulses through the medium containing two-level quantum particles with arbitrary dipole moments.
Considered in this paper the Maxwell-Bloch (MB) equations became known after Lamb [1-4]. In [5] Ablowitz, Kaup and Newell proposed the inverse scattering transform (IST) to the Maxwell-Bloch equations for studying a physical phenomenon known as the self-induced transparency. A description of general solutions to the MB equations and their classification was done in [6] by Gabitov, Zakharov and Mikhailov. In particular, they gave an approximate solution of the mixed problem to the MB equations in the domain $x,t\in(0,L)\times (0,\infty)$ and, on this bases, a description of the phenomenon of su…
We present exact N-soliton optical pulses riding on a continuous-wave (c.w.) beam that propagate through and interact with a two-level active optical medium. Their representation is derived via an appropriate generalization of the inverse scattering transform for the corresponding Maxwell-Bloch equations. We describe the single-soliton solutions in detail and classify them into several distinct families. In addition to the analogues of traveling-wave soliton pulses that arise in the absence of a c.w. beam, we obtain breather-like structures, periodic pulse-trains and rogue-wave-type (i.e., rat…
Recently, Zabolotskii [Phys. Rev. E 77, 036603 (2008)] presented the Lax pair for a version of the reduced Maxwell--Bloch equations. This version was derived under considering the unidirectional propagation of a two-component electromagnetic pulse through an anisotropic two-level medium. We demonstrate that his derivation contains essential omission, which led to the wrong version of the reduced Maxwell--Bloch equations. We also point out that the Lax pair for correct version of these equations had been known in the most general anisotropic case.
