New type of rogue waves
New type of localized solutions for the two-dimensional multicomponent Yajima-Oikawa system is presented. The dynamics of solutions of this type occurs on the zero background and is similar to that of rogue waves.
nlin.SI↗
arXiv subjects
N. V. Ustinov
Publications and source records attributed to N. V. Ustinov.
Appearing (disappearing) lumps and rogue lumps of the two-dimensional vector Yajima-Oikawa system
The solutions of the two-dimensional multicomponent Yajima-Oikawa system that have the functional arbitrariness are constructed by using the Darboux transformation technique. For the zero and constant backgrounds, different types of solutions of this system, including the lumps, line rogue waves, semi-rational solutions and their higher-order counterparts, are considered. Also, the generalization of the lump solutions (namely, appearing or disappearing lumps) is obtained in the two-component case under the special choice of the arbitrary functions. Then, the suitable ansatz is used to find the further generalization of these lumps (appearing-disappearing lumps or rogue lumps).
nlin.PS↗
Comment on 'Evolution of the unidirectional electromagnetic pulses in an anisotropic two-level medium'
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.
nlin.SI↗
New Kinds of Acoustic Solitons
We find that the modified sine-Gordon equation belonging to the class of the soliton equations describes the propagation of extremely short transverse acoustic pulses through the low-temperature crystal containing paramagnetic impurities with effective spin S=1/2 in the Voigt geometry case. The features of nonlinear dynamics of strain field and effective spins, which correspond to the different kinds of acoustic solitons, are studied.
nlin.SI↗
Optical Solitons in an Anisotropic Medium with Arbitrary Dipole Moments
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.
nlin.SI↗
Modes of nonlinear acoustic transparency in the strained paramagnetic crystal
The propagation of the transverse-longitudinal acoustic pulses through a strained cubic crystal containing the 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 the high-frequency and zero-frequency spectral components. We show that a pulse can propagate in the modes different from the acoustic self-induced transparency. In particular, a pulse propagating in the mode of 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.
nlin.PS↗
Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether's theorem
The hierarchy of the integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of the linearization of these equations and their conservation law in the terms of the solutions of the corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of the linearized equation due to the Noether's theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and the conservation laws that are explicitly expressed through the variables of the nonlinear equations are derived.
nlin.SI↗
Dynamics of two-component electromagnetic and acoustic extremely short pulses
The distinctive features of passing the two-component extremely short pulses through the nonlinear media are discussed. The equations considered describe the propagation in the two-level anisotropic medium of the electromagnetic pulses consisting of ordinary and extraordinary components and an evolution of the transverse-longitudinal acoustic pulses in a crystal containing the paramagnetic impurities with effective spin S=1/2. The solutions decreasing exponentially and algebraically are studied.
nlin.SI↗
Breather-like pulses in a medium with the permanent dipole moment
The solutions of the reduced Maxwell-Bloch equations for an anisotropic two-level medium, which describe the propagation of electromagnetic pulses having a duration from a few field oscillations, are studied. An influence of the permanent dipole moment of the quantum transition on dynamics of the pulses and their spectrum is considered.
nlin.SI↗
Optical transparency modes in anisotropic media
The modes of nonlinear propagation of the two-component electromagnetic pulses through optically uniaxial media containing resonant particles are studied. The features of their manifestation in the "dense" media and in the media with expressed positive and negative birefringences are discussed. It is shown that exponentially and rationally decreasing solutions of the system of material and wave equations allow us also to describe the propagation of the self-induced transparency pulses in isotropic media in the case, when the direct electric dipole-dipole interaction between the resonant particles is taken into account.
nlin.PS↗
The Lattice Equations of the Toda Type with an Interaction between a Few Neighborhoods
The sets of the integrable lattice equations, which generalize the Toda lattice, are considered. The hierarchies of the first integrals and infinitesimal symmetries are found. The properties of the multi-soliton solutions are discussed.
nlin.SI↗
Darboux Transformations, Infinitesimal Symmetries and Conservation Laws for Nonlocal Two-Dimensional Toda Lattice
The technique of Darboux transformation is applied to nonlocal partner of two-dimensional periodic A_{n-1} Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with quantized spectral parameter. The generalization of the Darboux transformation technique on linear equations of such a kind is given. The connections between the solutions of overdetermined linear systems and their expansions in series at singular points neighborhood are presented. The solutions of the nonlocal Toda lattice and infinite hierarchies of the infinitesimal symmetries and conservation laws are obtained.
nlin.SI↗
On Psi-function for finite-gap potentials
A way to derive an explicit formulae in terms of the potentials, if they are finite-gap, for the solutions of spectral problems and corresponding algebraic curves is presented.
nlin.SI↗
Darboux-integration of idρ/dt=[H,f(ρ)]
A Darboux-type method of solving the nonlinear von Neumann equation $i\dot ρ=[H,f(ρ)]$, with functions $f(ρ)$ commuting with $ρ$, is developed. The technique is based on a representation of the nonlinear equation by a compatibility condition for an overdetermined linear system. von Neumann equations with various nonlinearities $f(ρ)$ are found to possess the so-called self-scattering solutions. To illustrate the result we consider the Hamiltonian $H$ of a one-dimensional harmonic oscillator and $f(ρ)=ρ^q-2ρ^{q-1}$ with arbitary real $q$. It is shown that self-scattering solutions possess the same asymptotics for all $q$ and that different nonlinearities may lead to effectively indistinguishable evolutions. The result may have implications for nonextensive statistics and experimental tests of linearity of quantum mechanics.
quant-ph↗
On representation of the P-Q pair solution at the singular point neighborhood
The compatible expansion in series of solutions of both the equations of P-Q pair at neighborhood of the singular point is obtained in closed form for regular and irregular singularities. The conservation laws of the system of ordinary differential equations to arise from the compatibility condition of the P-Q pair are derived.
math-ph↗
Transformations of ordinary differential equations via Darboux transformation technique
A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of the solutions of the overdetermined linear systems are derived in the frameworks of the Darboux transformation technique.
math-ph↗
On exact solutions of Maxwell-Bloch system for two-level medium with degeneracy
Maxwell-Bloch system describing the resonant propagation of electromagnetic pulses in both two-level media with degeneracy in angle moment projection and three-level media with equal oscillator forces is considered. The inhomogeneous broadening of energy levels is accounted. Binary Darboux Transformation generating the solutions of the system is constructed. Pulses corresponding to the transition between levels with largest population difference are shown to be stable. The solution describing the propargation of pulses in the medium exited by the periodic wave is obtained. The hierarchy of infinitesimal symmetries is obtained by means of Darboux transformation.
quant-ph↗