Wave emission and absorption in a nonlinear integrable system

It is shown that in a nonlinear integrable system under specific conditions there occurs emission (or creation) and absorption (or annihilation) of waves. This phenomenon has essentially a nonlinear nature and is observed for the first time in the present nonlinear integrable system.     

Propagation of shape-preserving optical pulses in inhomogeneously broadened multi-level systems

We study stable propagation of multiple shape-preserving optical pulses in an inhomogeneously broadened multi-level atomic medium. By analytically solving the Maxwell-Schr?dinger equations governing the evolution of N coupled optical fields and atomic amplitudes we show that N pulsed optical waves coupling to (N+1)-levels can be automatically matched with the same soliton waveform and identical yet very slow propagation velocity. Several sets of coupled soliton solutions for two different (N+1)-level models are given and their stability is studied by using a numerical simulation.     

Six-component semi-discrete integrable nonlinear Schrödinger system

We suggest the six-component integrable nonlinear system on a quasi-one-dimensional lattice. Due to its symmetrical form, the general system permits a number of reductions; one of which treated as the semi-discrete integrable nonlinear Schrödinger system on a lattice with three structural elements in the unit cell is considered in considerable details. Besides six truly independent basic field variables, the system is characterized by four concomitant fields whose background values produce three additional types of inter-site resonant interactions between the basic fields. As a result, the system dynamics becomes associated with the highly nonstandard form of Poisson structure. The elementary Poisson brackets between all field variables are calculated and presented explicitly. The richness of system dynamics is demonstrated on the multi-component soliton solution written in terms of properly parameterized soliton characteristics.     

A nonlinear discrete integrable coupling system and its infinite conservation laws

We construct a nonlinear integrable coupling of discrete soliton hierarchy,and establish the infinite conservation laws(CLs) for the nonlinear integrable coupling of the lattice hierarchy.As an explicit application of the method proposed in the paper,the infinite conservation laws of the nonlinear integrable coupling of the Volterra lattice hierarchy are presented.     

Completely integrable models of nonlinear optics

The models of the nonlinear optics in which solitons appeared are considered. These models are of paramount importance in studies of nonlinear wave phenomena. The classical examples of phenomena of this kind are the self-focusing, self-induced transparency and parametric interaction of three waves. At present there are a number of theories based on completely integrable systems of equations, which are, both, generations of the original known models and new ones. The modified Korteweg-de Vries equation, the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equation. Sine-Gordon equation, the reduced Maxwell-Bloch equation. Hirota equation, the principal chiral field equations, and the equations of massive Thirring model are some soliton equations, which are usually to be found in nonlinear optics theory.     

Semi-discrete integrable nonlinear Schrödinger system with background-controlled inter-site resonant coupling

We summarize the most featured items characterizing the semi-discrete nonlinear Schrödinger system with background-controlled inter-site resonant coupling. The system is shown to be integrable in the Lax sense that make it possible to obtain its soliton solutions in the framework of properly parameterized dressing procedure based on the Darboux transformation. On the other hand the system integrability inspires an infinite hierarchy of local conservation laws some of which were found explicitly in the framework of generalized recursive approach. The system consists of two basic dynamic subsystems and one concomitant subsystem and it permits the Hamiltonian formulation accompanied by the highly nonstandard Poisson structure. The nonzero background level of concomitant fields mediates the appearance of an additional type of inter-site resonant coupling and as a consequence it establishes the triangular-lattice-ribbon spatial arrangement of location sites for the basic field excitations. Adjusting the background parameter we are able to switch over the system dynamics between two essentially different regimes separated by the critical point. The system criticality against the background parameter is manifested both indirectly by the auxiliary linear spectral problem and directly by the nonlinear dynamical equations themselves. The physical implications of system criticality become evident after the rather sophisticated canonization procedure of basic field variables. There are two variants of system standardization equal in their rights. Each variant is realizable in the form of two nonequivalent canonical subsystems. The broken symmetry between canonical subsystems gives rise to the crossover effect in the nature of excited states. Thus in the under-critical region the system support the bright excitations in both subsystems, while in the over-critical region one of subsystems converts into the subsystem of dark excitations.     

Prolongation structure of a new integrable system

We have obtained the inverse scattering equations associated with a new pair of coupled nonlinear evolution equations in two dimensions. The spectral parameter is introduced by invoking the invariance of the equation set, and imposing those on the Lax pair.     

Theory of integrable three-dimensional nonlinear lattice equations

Letters in Mathematical Physics - A linear inhomogeneous system of equations, which is invariant under a general type of nonlocal integral transformations, is investigated. The compatibility...     

Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy

In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. From the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.     

Soliton-like propagation of light pulses in a nonlinear photonic crystal

On the basis of computer simulation, we analyze the dynamics of interaction of subpulses localized inside certain layers of a one-dimensional nonlinear photonic crystal. Localized subpulses are shown to move and interact like solitons. Possible scenarios of their interaction are described: a spatially periodic regime, the merging of several subpulses with similar characteristics into a single higher intensity and more slowly propagating subpulse, and the formation of an immovable pulse in a layer of the photonic crystal. To verify the results of the computer simulation and to elucidate the nature of spatially localized structures forming, a comparison with the analytical solution for a medium with cubic nonlinearity is performed.     

Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy

In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. From the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.     

Capture and confinement of solitons in nonlinear integrable systems

The system describing the interaction of a long wave with a short wave packet on thex,y plane is considered. The solutions are found to describe a soliton that comes from infinity and then is captured into a conditionally periodical oscillatory regime. The solutions are also found that describe a soliton coming from infinity and then decaying into two solitons: one goes to infinity and the other is captured into a conditionally periodical oscillatory regime. The obtained results are relevant to some problems of hydrodynamics, plasma physics, solid state physics, etc.     

On the stability of nonlinear waves in integrable models

A new method of stability investigation is presented for solutions of nonlinear equations integrable with the help of the inverse scattering transform (IST). The stability problem for periodic nonlinear waves in weakly dispersive media is solved with respect to transverse perturbations. It is shown that for positive dispersion media one-dimensional waves are unstable, and for negative dispersion such waves are stable.     

Propagation of ultrashort pulses in a nonlinear long-period fiber grating

By solving a pair of normalized nonlinear coupled-mode equations, we analyze in detail the propagation of ultrashort optical pulses in a nonlinear long-period fiber grating and investigate the conditions for the grating to function as an optical limiter or a saturable absorber. We show that the function of the grating depends on whether the nonlinear effect counteracts or enhances the detuning effect, and the nonlinear effect can be weakened significantly by the group-delay difference between the core mode and the coupled cladding mode. We present simulation results to illustrate these effects and discuss the physical conditions required for an effective operation of a nonlinear long-period grating.     

Generation of unipolar pulses in nonlinear media

Methods recently proposed for generating unipolar pulses in nonlinear media in terahertz and optical electromagnetic ranges are reviewed. Such pulses have nonzero “electric area” (time integral of the field strength over the entire duration of a pulse) and, correspondingly, a significant component of the field with zero frequency, thus exhibiting quasistatic properties. Effective generation of unipolar pulses would allow, e.g., transferring mechanical momentum to charged particles and, thereby, controlling the motion of wave packets of matter, which can be useful for compact accelerators of charged particles and for other applications.     

On the solutions in elliptic functions of integrable nonlinear equations

A new solution in elliptic functions for the KdV equation is constructed using the method proposed by Belokolos and the author.     

Collisions of unipolar subcycle pulses in a nonlinear resonantly absorbing medium

Collisions of unipolar subcycle pulses in a nonlinear resonant medium that coherently interact with it are studied theoretically. The dynamics of spatial polarization structures and population difference that the pulses induce in the medium are analyzed. A surprising feature is that the medium is capable of “remembering” the result of the interaction with the pulses and their collisions during times comparable to the polarization relaxation time. An analysis of the dynamics of light-induced structures makes it possible to judge the parameters of subcycle pulses at the times longer than the pulse duration, which, in the future, can be useful in their detection.     

Propagation of ultrashort pulses in a nonlinear two-core photonic crystal fiber

We analyze the propagation of ultrashort pulses in a nonlinear two-core photonic crystal fiber (PCF) by solving a pair of coupled-mode equations that include all the significant linear and nonlinear terms. In particular, we highlight the fact that the coupling coefficient dispersion can cause significant pulse distortion over a short length of a two-core PCF. We also study all-optical switching and multi-frequency generation and obtain a reasonable agreement with recent experimental data.