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I. V. Barashenkov

Publications and source records attributed to I. V. Barashenkov.

At least 19 recordsLinked to original sources
arXiv2026

Swinging Waves in the Ablowitz-Ladik Equation

We construct a novel family of exact cnoidal wave and soliton solutions of the focusing and defocusing Ablowitz-Ladik equations. Unlike cnoidal waves that were obtained by earlier authors, the phase variable of the new solutions exhibits a nonlinear dependence on time and site number; the wave ``swings". Our approach hinges on the existence of a two-point map governing the absolute value of the complex field; this map gives rise to standing waves centred arbitrarily relative to the lattice sites. Having derived stationary solutions, we use these as a basis for constructing waves with nonzero velocity. The localised members of the new family comprise dark solitons with the nontrivial asymptotic behaviour. We identify periodic and quasiperiodic patterns and establish an explicit quantisation rule for the velocity of the wave circulating in a closed loop of $N$ sites.

nlin.PS↗
arXiv2024

The energy-frequency diagram of the (1+1)-dimensional $Φ^4$ oscillon

Two different methods are used to study the existence and stability of the (1+1)-dimensional $Φ^4$ oscillon. The variational technique approximates it by a periodic function with a set of adiabatically changing parameters. An alternative approach treats oscillons as standing waves in a finite-size box; these are sought as solutions of a boundary-value problem on a two-dimensional domain. The numerical analysis reveals that the standing wave's energy-frequency diagram is fragmented into disjoint segments with $ω_{n+1} < ω< ω_{n}$, where $ω_n= ω_0/ (n+1)$, $n=0,1,2, ...$, and $ω_0$ is the endpoint of the continuous spectrum (mass threshold of the model). The variational approximation involving the first, zeroth and second harmonic components provides an accurate description of the oscillon with the frequency in $(ω_1, ω_0)$, but breaks down as $ω$ falls out of that interval.

hep-th↗
arXiv2023

Variational formalism for the Klein-Gordon oscillon

The variational method employing the amplitude and width as collective coordinates of the Klein-Gordon oscillon leads to a dynamical system with unstable periodic orbits that blow up when perturbed. We propose a multiscale variational approach free from the blow-up singularities. An essential feature of the proposed trial function is the inclusion of the third collective variable: a correction for the nonuniform phase growth. In addition to determining the parameters of the oscillon, our approach detects the onset of its instability.

hep-th↗
arXiv2023

Understanding oscillons: standing waves in a ball

Oscillons are localised long-lived pulsating states in the three-dimensional $ϕ^4$ theory. We gain insight into the spatio-temporal structure and bifurcation of the oscillons by studying time-periodic solutions in a ball of a finite radius. A sequence of weakly localised {\it Bessel waves} -- nonlinear standing waves with the Bessel-like $r$-dependence -- is shown to extend from eigenfunctions of the linearised operator. The lowest-frequency Bessel wave serves as a starting point of a branch of periodic solutions with exponentially localised cores and small-amplitude tails decaying slowly towards the surface of the ball. A numerical continuation of this branch gives rise to the energy-frequency diagram featuring a series of resonant spikes. We show that the standing waves associated with the resonances are born in the period-multiplication bifurcations of the Bessel waves with higher frequencies. The energy-frequency diagram for a sufficiently large ball displays sizeable intervals of stability against spherically-symmetric perturbations.

hep-th↗
arXiv2021

Gyrating solitons in a necklace of optical waveguides

We consider light pulses in a circular array of $2N$ coupled nonlinear optical waveguides. The waveguides are either hermitian or alternate gain and loss in a $\mathcal{PT}$-symmetric fashion. Simple patterns in the array include a ring of $2N$ pulses travelling abreast, and a breather -- a string of pulses where all even and all odd waveguides flash in turn. In addition, the structure displays solitons gyrating around the necklace by switching from one waveguide to the next. Some of the gyrating solitons are stable while other ones are weakly unstable and evolve into gyrating multiflash strings. By tuning the gain-loss coefficient, the gyration of solitons in a nonhermitian array may be reversed without changing the direction of their translational motion.

physics.optics↗
arXiv2020

Stable solitons in a nearly PT-symmetric ferromagnet with spin-transfer torque

We consider the Landau-Lifshitz equation for the spin torque oscillator - a uniaxial ferromagnet in an external magnetic field with polarised spin current driven through it. In the absence of the Gilbert damping, the equation turns out to be PT-symmetric. We interpret the PT-symmetry as a balance between gain and loss - and identify the gaining and losing modes. In the vicinity of the bifurcation point of a uniform static state of magnetisation, the PT-symmetric Landau-Lifshitz equation with a small dissipative perturbation reduces to a nonlinear Schrödinger equation with a quadratic nonlinearity. The analysis of the Schrödinger dynamics demonstrates that the spin torque oscillator supports stable magnetic solitons. The PT near-symmetry is crucial for the soliton stability: the addition of a finite dissipative term to the Landau-Lifshitz equation destabilises all solitons that we have found.

nlin.PS↗
arXiv2019

Global search for localised modes in scalar and vector nonlinear Schrödinger-type equations

We present a new approach for search of coexisting classes of localised modes admitted by the repulsive (defocusing) scalar or vector nonlinear Schrödinger-type equations. The approach is based on the observation that generic solutions of the corresponding stationary system have singularities at finite points on the real axis. We start with establishing conditions on the initial data of the associated Cauchy problem that guarantee the formation of a singularity. Making use of these sufficient conditions, we identify the bounded, nonsingular, solutions --- and then classify them according to their asymptotic behaviour. To determine the bounded solutions, a properly chosen space of initial data is scanned numerically. Due to asymptotic or symmetry considerations, we can limit ourselves to a one- or two-dimensional space. For each set of initial conditions we compute the distances $X^{\pm}$ to the nearest forward and backward singularities; large $X^+$ or $X^-$ indicate the proximity to a bounded solution. We illustrate our method with the Gross-Pitaevskii equation with a $\PT$-symmetric complex potential, a system of coupled Gross-Pitaevskii equations with real potentials, and the Lugiato-Lefever equation with normal dispersion.

nlin.PS↗
arXiv2018

Spinor solitons and their $\mathcal{PT}$-symmetric offspring

Although the spinor field in (1+1) dimensions has the right structure to model a dispersive bimodal system with gain and loss, the plain addition of gain to one component of the field and loss to the other one results in an unstable dispersion relation. In this paper, we advocate a different recipe for the $\mathcal{PT}$-symmetric extension of spinor models --- the recipe that does not produce instability of the linear Dirac equation. Having exemplified the physical origins of the $\mathcal P$- and $\mathcal T$-breaking terms, we consider the extensions of three U(1)-invariant spinor models with cubic nonlinearity. Of these, the \PT-symmetric extension of the Thirring model is shown to be completely integrable and possess infinitely many conserved quantities. The \PT-symmetric Gross-Neveu equation conserves energy and momentum but does not conserve charge. The third model is introduced for the purpose of comparison with the previous two; its \PT-symmetric extension has no conservation laws at all. Despite this dramatic difference in the integrability and conservation properties, all three \PT-symmetric models are shown to have exact soliton solutions. Similar to the solitons of the extended Thirring and Gross-Neveu equations, the solitons of the new model are found to be stable --- except for a narrow band of frequencies adjacent to the soliton existence boundary. The persistence under the $\mathcal P$- and $\mathcal T$-breaking perturbations as well as the prevalence of stability highlight a remarkable sturdiness of spinor solitons in (1+1) dimensions.

math-ph↗
arXiv2016

Stationary through-flows in a Bose-Einstein condensate with a PT-symmetric impurity

Superfluid currents in the boson condensate with a source and sink of particles are modelled by the PT-symmetric Gross-Pitaevskii equation with a complex potential. We demonstrate the existence of through-flows of the condensate --- stationary states with the asymptotically nonvanishing flux. The through-flows come in two broad varieties determined by the form of their number density distribution. One variety is described by dip-like solutions featuring a localised density depression; the other one comprises hump-like structures with a density spike in their core. We exemplify each class by exact closed-form solutions. For a fixed set of parameters of the PT-symmetric potential, stationary through-flows form continuous families parametrized by the strength of the background flux. All hump-like and some dip-like members of the family are found to be stable. We show that the through-flows can be controlled by varying the gain-and-loss amplitude of the complex potential and that these amplitude variations may produce an anomalous response of the flux across the gain-loss interface.

cond-mat.quant-gas↗
arXiv2016

Jamming anomaly in $\mathcal{PT}$-symmetric systems

The Schrödinger equation with a $\mathcal{PT}$-symmetric potential is used to model an optical structure consisting of an element with gain coupled to an element with loss. At low gain-loss amplitudes $γ$, raising the amplitude results in the energy flux from the active to the leaky element being boosted. We study the anomalous behaviour occurring for larger $γ$, where the increase of the amplitude produces a drop of the flux across the gain-loss interface. We show that this jamming anomaly is either a precursor of the exceptional point, where two real eigenvalues coalesce and acquire imaginary parts, or precedes the eigenvalue's immersion in the continuous spectrum.

physics.optics↗
arXiv2015

Localised nonlinear modes in the PT-symmetric double-delta well Gross-Pitaevskii equation

We construct exact localised solutions of the PT-symmetric Gross-Pitaevskii equation with an attractive cubic nonlinearity. The trapping potential has the form of two $δ$-function wells, where one well loses particles while the other one is fed with atoms at an equal rate. The parameters of the constructed solutions are expressible in terms of the roots of a system of two transcendental algebraic equations. We also furnish a simple analytical treatment of the linear Schrödinger equation with the PT-symmetric double-$δ$ potential.

nlin.PS↗
arXiv2015

Exactly solvable Wadati potentials in the PT-symmetric Gross-Pitaevskii equation

This note examines Gross-Pitaevskii equations with PT-symmetric potentials of the Wadati type: $V=-W^2+iW_x$. We formulate a recipe for the construction of Wadati potentials supporting exact localised solutions. The general procedure is exemplified by equations with attractive and repulsive cubic nonlinearity bearing a variety of bright and dark solitons.

nlin.PS↗
arXiv2015

Dimer with gain and loss: Integrability and $\mathcal{PT}$-symmetry restoration

A $\mathcal{PT}$-symmetric nonlinear Schrödinger dimer is a two-site discrete nonlinear Schrödinger equation with one site losing and the other one gaining energy at the same rate. In this paper, two four-parameter families of cubic $\mathcal{PT}$-symmetric dimers are constructed as gain-loss extensions of their conservative, Hamiltonian, counterparts. We prove that all these damped-driven equations define completely integrable Hamiltonian systems. The second aim of our study is to identify nonlinearities that give rise to the spontaneous $\mathcal{PT}$-symmetry restoration. When the symmetry of the underlying linear dimer is broken and an unstable small perturbation starts to grow, the nonlinear coupling of the required type diverts progressively large amounts of energy from the gaining to the losing site. As a result, the exponential growth is saturated and all trajectories remain trapped in a finite part of the phase space regardless of the value of the gain-loss coefficient.

nlin.SI↗
arXiv2013

Actively coupled optical waveguides

We consider light propagation through a pair of nonlinear optical waveguides with absorption, placed in a medium with power gain. The active medium boosts the in-phase component of the overlapping evanescent fields of the guides, while the nonlinearity of the guides couples it to the damped out-of-phase component creating a feedback loop. As a result, the structure exhibits stable stationary and oscillatory regimes in a wide range of gain-loss ratios. We show that the pair of actively-coupled ($\mathcal{AC}$) waveguides can act as a stationary or integrate-and-fire comparator sensitive to tiny differences in their input powers.

physics.optics↗
arXiv2012

Breathers in PT-symmetric optical couplers

We show that the parity-time (PT) symmetric coupled optical waveguides with gain and loss support localised oscillatory structures similar to the breathers of the classical $ϕ^4$ model. The power carried by the PT-breather oscillates periodically, switching back and forth between the waveguides, so that the gain and loss are compensated on the average. The breathers are found to coexist with solitons and be prevalent in the products of the soliton collisions. We demonstrate that the evolution of the small-amplitude breather's envelope is governed by a system of two coupled nonlinear Schrödinger equations, and employ this Hamiltonian system to show that the small-amplitude PT-breathers are stable.

nlin.PS↗
arXiv2012

Optical solitons in $\mathcal{PT}$-symmetric nonlinear couplers with gain and loss

We study spatial and temporal solitons in the $\mathcal{PT}$ symmetric coupler with gain in one waveguide and loss in the other. Stability properties of the high- and low-frequency solitons are found to be completely determined by a single combination of the soliton's amplitude and the gain/loss coefficient of the waveguides. The unstable perturbations of the high-frequency soliton break the symmetry between its active and lossy components which results in a blowup of the soliton or a formation of a long-lived breather state. The unstable perturbations of the low-frequency soliton separate its two components in space blocking the power drainage of the active component and cutting the power supply to the lossy one. Eventually this also leads to the blowup or breathing.

nlin.PS↗
arXiv2011

Stability of cnoidal waves in the parametrically driven nonlinear Schrödinger equation

The parametrically driven, damped nonlinear Schrödinger equation has two cn- and two dn-wave solutions. We show that one pair of the cn and dn solutions is unstable for any combination of the driver's strength, dissipation coefficient and spatial period of the wave; this instability is against periodic perturbations. The second dn-wave solution is shown to be unstable against antiperiodic perturbations --- in a certain region of the parameter space. We also consider quasiperiodic perturbations with long modulation wavelength, in the limit where the driving strength is only weakly exceeding the damping coefficient.

nlin.PS↗