arXiv subjects
Publications and source records attributed to V. V. Konotop.
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.
It is demonstrated that a two-component Bose-Einstein condensate (BEC) with all-repulsive inter-atomic interactions loaded into a radially symmetric harmonic trap supports robust non-coaxial vortices with approximately orthogonal vortex lines in each of the components. These cross vortices are excited from the linear modes by a sudden switch-on of the nonlinearity (via Feshbach resonance) and are characterized by persistent dynamical regimes of precession with nutation, resembling the motion of a rigid body. The obtained dynamics can be understood qualitatively on the basis of a simple mechanical model.
The dynamics of two component coupled Airy beams is investigated. In the linear propagation regime a complete analytic solution describes breather like propagation of the two components featuring non-diffracting self-accelerating Airy behavior. The superposition of two beams with different input properties opens the possibility to design more complex non-diffracting propagation scenarios. In the strongly nonlinear regime the dynamics remains qualitatively robust as is revealed by direct numerical simulations. Due to the Kerr effect the two beams emit solitonic breathers, whose coupling period is compatible with the remaining Airy-like beams. The results of this study are relevant for the description of photonic and plasmonic beams propagating in coupled planar waveguides as well as for birefrigent or multi-wavelengths beams.
We introduce the notion of a ${\cal PT}$-symmetric dimer with a $χ^{(2)}$ nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity should be accessible in a pair of optical waveguides with quadratic nonlinearity and gain and loss, respectively. An interesting feature of the problem is that because of the two harmonics, there exist in general two distinct gain/loss parameters, different values of which are considered herein. We find a number of traits that appear to be absent in the more standard cubic case. For instance, bifurcations of nonlinear modes from the linear solutions occur in two different ways depending on whether the first or the second harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover, a host of interesting bifurcation phenomena appear to occur including saddle-center and pitchfork bifurcations which our parametric variations elucidate. The existence and stability analysis of the stationary solutions is corroborated by numerical time-evolution simulations exploring the evolution of the different configurations, when unstable.
Stable discrete compactons in arrays of inter-connected three-line waveguide arrays are found in linear and nonlinear limits in conservative and in parity-time PT symmetric models. The compactons result from the interference of the fields in the two lines of waveguides ensuring that the third (middle) line caries no energy. PT-symmetric compactons require not only the presence of gain and losses in the two lines of the waveguides but also complex coupling, i.e. gain and losses in the coupling between the lines carrying the energy and the third line with zero field. The obtained compactons can be stable and their branches can cross the branches of the dissipative solitons. Unusual bifurcations of branches of solitons from linear compactons are described.
We study the existence and stability of solitons in the quadratic nonlinear media with spatially localized ${\cal PT}$-symmetric modulation of the linear refractive index. Families of stable one and two hump solitons are found. The properties of nonlinear modes bifurcating from a linear limit of small fundamental harmonic field are investigated. It is shown that the fundamental branch, bifurcating from the linear mode of the fundamental harmonic is limited in power. The power maximum decreases with the strength of the imaginary part of the refractive index. The modes bifurcating from the linear mode of the second harmonic can exist even above ${\cal PT}$ symmetry breaking threshold. We found that the fundamental branch bifurcating from the linear limit can undergo a secondary bifurcation colliding with a branch of two-hump soliton solutions. The stability intervals for different values of the propagation constant and gain/loss gradient are obtained. The examples of dynamics and excitations of solitons obtained by numerical simulations are also given.
We propose that parametrically excited patterns, also known as Faraday patterns, can be observed in nonlinear fiber resonators, where the coefficient of Kerr nonlinearity is periodically varying along the fiber in resonator. We study the parametric instability analytically on the basis of the Floquet theory, also numerically, by direct integration of the system. Instead of classical Faraday wave excitation scenario, where modulation in time causes formation of patterns in space, here we propose an inverted scenario, where the modulation in space excites the patterns in time.
We consider a binary Bose-Einstein condensate (BEC) with nonlinear repulsive interactions and linear spin-orbit (SO) and Zeeman-splitting couplings. In the presence of the trapping harmonic-oscillator (HO) potential, we report the existence of even, odd, and asymmetric spatial modes. They feature alternating domains with opposite directions of the pseudo-spin, i.e., anti-ferromagnetic structures, which is explained by the interplay of the linear couplings, HO confinement, and repulsive self-interaction. The number of the domains is determined by the strength of the SO coupling. The modes are constructed analytically in the weakly nonlinear system. The dynamical stability of the modes is investigated by means of the Bogoliubov-de Gennes equations and direct simulations. A notable result is that the multi-domain-wall (DW) structures are stable, alternating between odd and even shapes, while the simplest single-DW structure is unstable. Thus, the system features a transition to the complex ground states under the action of the SO coupling. The addition of the Zeeman splitting transforms the odd modes into asymmetric ones via spontaneous symmetry breaking. The results suggest possibilities for switching the binary system between states with opposite (pseudo) magnetization by external fields, and realization of similar stable states and dynamical effects in solid-state and nonlinear-optical settings emulated by the SO-coupled BECs.
We considered the modulational instability of continuous-wave backgrounds, and the related generation and evolution of deterministic rogue waves in the recently introduced parity-time (PT)-symmetric system of linearly-coupled nonlinear Schrödinger equations, which describes a Kerr-nonlinear optical coupler with mutually balanced gain and loss in its cores. Besides the linear coupling, the overlapping cores are coupled through cross-phase-modulation term too. While the rogue waves, built according to the pattern of the Peregrine soliton, are (quite naturally) unstable, we demonstrate that the focusing cross-phase-modulation interaction results in their partial stabilization. For PT-symmetric and antisymmetric bright solitons, the stability region is found too, in an exact analytical form, and verified by means of direct simulations.
We investigate, both analytically and numerically, the quasi-superfluidity properties of periodic Bose-Einstein condensates (BECs) in a quasi-one-dimensional (1D) ring with optical lattices (OL) of different kinds (linear and nonlinear) and with a moving defect of an infinite mass inside. To study the dynamics of the condensate we used a mean-field approximation describing the condensate by use of the Gross-Pitaevskii equation for the order parameter. We show that the resonant scattering of sound Bloch waves with the defect profoundly affect BEC superfluidity. In particular, a moving defect always leads to the breakdown of superfluidity independently of the value of its velocity. For weak periodic potentials the superfluidity breakdown may occur on a very long time scale (quasisuperfluidity) but the breakdown process can be accelerated by increasing the strength of the OL. Quite remarkably, we find that when the length of the ring is small enough to imply the discreteness of the reciprocal space, it becomes possible to avoid the resonant scattering and to restore quasi-superfluidity.
In this work, we propose a PT-symmetric coupler whose arms are birefringent waveguides as a realistic physical model which leads to a so-called quadrimer i.e., a four complex field setting. We seek stationary solutions of the resulting linear and nonlinear model, identifying its linear point of PT symmetry breaking and examining the corresponding nonlinear solutions that persist up to this point, as well as, so-called, ghost states that bifurcate from them. We obtain the relevant symmetry breaking bifurcations and numerically follow the associated dynamics which give rise to growth/decay even within the PT-symmetric phase. Our obtained stationary nonlinear solutions are found to terminate in saddle-center bifurcations which are analogous to the linear PT-phase transition.
We prove existence of discrete solitons in infinite parity-time (PT-) symmetric lattices by means of analytical continuation from the anticontinuum limit. The energy balance between dissipation and gain implies that in the anticontinuum limit the solitons are constructed from elementary PT-symmetric blocks such as dimers, quadrimers, or more general oligomers. We consider in detail a chain of coupled dimers, analyze bifurcations of discrete solitons from the anticontinuum limit and show that the solitons are stable in a sufficiently large region of the lattice parameters. The generalization of the approach is illustrated on two examples of networks of quadrimers, for which stable discrete solitons are also found.
We theoretically demonstrate the possibility to observe the macroscopic Zeno effect for nonlinear waveguides with a localized dissipation. We show the existence of stable stationary flows, which are balanced by the losses in the dissipative domain. The macroscopic Zeno effect manifests itself in the non-monotonic dependence of the stationary flow on the strength of the dissipation. In particular, we highlight the importance of the parameters of the dissipation to observe the phenomenon. Our results are applicable to a large variety of systems, including condensates of atoms or quasi-particles and optical waveguides.
By rearrangements of waveguide arrays with gain and losses one can simulate transformations among parity-time (PT-) symmetric systems not affecting their pure real linear spectra. Subject to such transformations, however, the nonlinear properties of the systems undergo significant changes. On an example of an array of four waveguides described by the discrete nonlinear Schrödinger equation with dissipation and gain, we show that the equivalence of the underlying linear spectra implies similarity of neither structure nor stability of the nonlinear modes in the arrays. Even the existence of one-parametric families of nonlinear modes is not guaranteed by the PT symmetry of a newly obtained system. Neither the stability is directly related to the PT symmetry: stable nonlinear modes exist even when the spectrum of the linear array is not purely real. We use graph representation of PT-symmetric networks allowing for simple illustration of linearly equivalent networks and indicating on their possible experimental design.
Superfluid properties of Bose-Einstein condensates (BEC) in toroidal quasi-one-dimensional traps are investigated in the presence of periodic scattering length modulations along the ring. The existence of several types of stable periodic waves, ranging from almost uniform to very fragmented chains of weakly interacting and equally spaced solitons, is demonstrated. We show that these waves may support persistent atomic currents and sound waves with spectra of Bogoliubov type. Fragmented condensates can be viewed as arrays of Josephson junctions and the current as a BEC manifestation of the dc-Josephson effect. The influence of linear defects on BEC superfluidity has been also investigated. We found that for subcritical velocities, linear defects that are static with respect to the lattice (while the condensate moves in respect to both the optical lattice and the defect) preserve the BEC superfluidity.
We report a novel algorithm of constructing linear and nonlinear potentials in the two-dimensional Gross-Pitaevskii equation subject to given boundary conditions, which allow for exact analytic solutions. The obtained solutions represent superfluid flows in inhomogeneous Bose-Einstein condensates. The method is based on the combination of the similarity reduction of the two-dimensional Gross-Pitaevskii equation to the one-dimensional nonlinear Schrodinger equation, the latter allowing for exact solutions, with the conformal mapping of the given domain, where the flow is considered, to a half-space. The stability of the obtained flows is addressed. A number of stable and physically relevant examples are described.
The Zeno effect is investigated for soliton type pulses in a nonlinear directional coupler with dissipation. The effect consists in increase of the coupler transparency with increase of the dissipative losses in one of the arms. It is shown that localized dissipation can lead to switching of solitons between the arms. Power losses accompanying the switching can be fully compensated by using a combination of dissipative and active (in particular, parity-time -symmetric) segments.
We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT) symmetric potential. We are particularly focusing on the case where the spatially-dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and $\tanh$-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, what suggests that the relation between width of the modes and spatial size of the complex potential define the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.