arXiv subjects

I V Barashenkov

Publications and source records attributed to I V Barashenkov.

8 recordsLinked to original sources

arXiv2023

Integrability and trajectory confinement in $\mathcal{PT}$-symmetric waveguide arrays

We consider $\mathcal{PT}$-symmetric ring-like arrays of optical waveguides with purely nonlinear gain and loss. Regardless of the value of the gain-loss coefficient, these systems are protected from spontaneous $\mathcal{PT}$-symmetry breaking. If the nonhermitian part of the array matrix has cross-compensating structure, the total power in such a system remains bounded -- or even constant -- at all times. We identify two-, three-, and four-waveguide arrays with cross-compensatory nonlinear gain and loss that constitute completely integrable Hamiltonian systems.

physics.optics↗

arXiv2018

The Continuing Story of the Wobbling Kink

The wobbling kink is the soliton of the $ϕ^4$ model with an excited internal mode. We outline an asymptotic construction of this particle-like solution that takes into account the coexistence of several space and time scales. The breakdown of the asymptotic expansion at large distances is prevented by introducing the long-range variables "untied" from the short-range oscillations. We formulate a quantitative theory for the fading of the kink's wobbling due to the second-harmonic radiation, explain the wobbling mode's longevity and discuss ways to compensate the radiation losses. The compensation is achieved by the spatially uniform driving of the kink, external or parametric, at a variety of resonant frequencies. For the given value of the driving strength, the largest amplitude of the kink's oscillations is sustained by the {\it parametric\/} pumping --- at its natural wobbling frequency. This type of forcing also produces the widest Arnold tongue in the "driving strength versus driving frequency" parameter plane. As for the {\it external\/} driver with the same frequency, it brings about an interesting rack and pinion mechanism that converts the energy of external oscillation to the translational motion of the kink.

nlin.PS↗

arXiv2017

Solitons in PT-symmetric ladders of optical waveguides

We consider a PT-symmetric ladder-shaped optical array consisting of a chain of waveguides with gain coupled to a parallel chain of waveguides with loss. All waveguides have the focusing Kerr nonlinearity. The array supports two co-existing solitons, an in-phase and an antiphase one, and each of these can be centred either on a lattice site or midway between two neighbouring sites. We show that both bond-centred (i.e. intersite) solitons are unstable regardless of their amplitudes and parameters of the chain. The site-centred in-phase soliton is stable when its amplitude lies below a threshold that depends on the coupling and gain-loss coefficient. The threshold is lowest when the gain-to-gain and loss-to-loss coupling constant in each chain is close to the interchain gain-to-loss coupling coefficient. The antiphase soliton in the strongly-coupled chain or in a chain close to the PT-symmetry breaking point, is stable when its amplitude lies above a critical value and unstable otherwise. The instability growth rate of solitons with small amplitude is exponentially small in this parameter regime; hence the small-amplitude solitons, though unstable, have exponentially long lifetimes. On the other hand, the antiphase soliton in the weakly or moderately coupled chain and away from the PT-symmetry breaking point, is unstable when its amplitude falls in one or two finite bands. All amplitudes outside those bands are stable.

nlin.PS↗

arXiv2014

An exactly solvable $\mathcal{PT}$-symmetric dimer from a Hamiltonian system of nonlinear oscillators with gain and loss

We show that a pair of coupled nonlinear oscillators, of which one oscillator has positive and the other one negative damping of equal rate, can form a Hamiltonian system. Small-amplitude oscillations in this system are governed by a $\mathcal{PT}$-symmetric nonlinear Schrödinger dimer with linear and cubic coupling. The dimer also represents a Hamiltonian system and is found to be exactly solvable in elementary functions. We show that the nonlinearity softens the $\mathcal{PT}$-symmetry breaking transition in the nonlinearly-coupled dimer: stable periodic and quasiperiodic states with large enough amplitudes persist for an arbitrarily large value of the gain-loss coefficient.

nlin.SI↗

arXiv2013

Blow-up regimes in the $\mathcal{PT}$- and the $\mathcal{AC}$-dimer

In the actively coupled ($\mathcal{AC}$) pair of waveguides, the growth of small perturbations is saturated by the focussing nonlinearity that couples the linearly growing to the linearly damped mode. On the other hand, in the $\mathcal{PT}$-symmetric coupler, the focussing nonlinearity promotes the blowup of stationary light beams. The purpose of this study is to compare the nonlinear dynamics and explain the opposite effect of the same nonlinearity in the two systems. We show that while the blowup regimes are stable in the $\mathcal{PT}$-symmetric pair of waveguides, they are unstable and hence cannot be observed in the $\mathcal{AC}$-dimer.

physics.optics↗

arXiv2013

$\mathcal{PT}$-symmetry breaking in a necklace of coupled optical waveguides

We consider parity-time ($\mathcal{PT}$) symmetric arrays formed by $N$ optical waveguides with gain and $N$ waveguides with loss. When the gain-loss coefficient exceeds a critical value $γ_c$, the $\mathcal{PT}$-symmetry becomes spontaneously broken. We calculate $γ_c(N)$ and prove that $γ_c \to 0$ as $N \to \infty$. In the symmetric phase, the periodic array is shown to support $2N$ solitons with different frequencies and polarisations.

physics.optics↗

arXiv2008

Asymptotic expansion of the wobbling kink

The method of multiple scales is used to study the wobbling kink of the $ϕ^4$ equation. The amplitude of the wobbling is shown to decay very slowly, as $t^{-1/2}$, and hence the wobbler turns out to be an extremely long-lived object.

nlin.PS↗

arXiv2008

Stability of the Bloch wall via the Bogomolnyi decomposition in elliptic coordinates

We consider the one-dimensional anisotropic XY model in the continuum limit. Stability analysis of its Bloch wall solution is hindered by the nondiagonality of the associated linearised operator and the hessian of energy. We circumvent this difficulty by showing that the energy admits a Bogomolnyi bound in elliptic coordinates and that the Bloch wall saturates it -- that is, the Bloch wall renders the energy minimum. Our analysis provides a simple but nontrivial application of the BPS (Bogomolnyi - Prasad - Sommerfield) construction in one dimension, where its use is often believed to be limited to reproducing results obtainable by other means.

nlin.PS↗