Nonlinear Stationary States Research Articles

Stationary states and dynamic regimes for a nonlinear asymmetric trimer with gain and loss

The paper studies a model of three couplers with Kerr-type nonlinearity in an asymmetric form with gain and loss. Analytical solutions of nonlinear stationary states are constructed in the vicinity of the neutral stability parameters for the null solution. The stability of such states is investigated analytically. Families of nonlinear stationary states depending on the gain coefficient are built numerically. The study of stationary solutions has revealed the parameters of oscillating and quasi-periodic dynamic regimes, which turn out to be attractors for a wide variety of initial data. The investigated properties may be useful for creating optical processing devices.

Stability and tip streaming of a surfactant-loaded drop in an extensional flow. Influence of surface viscosity

We study numerically the nonlinear stationary states of a droplet covered with an insoluble surfactant in a uniaxial extensional flow. We calculate both the eigenfunctions to reveal the instability mechanism and the time-dependent states resulting from it, which provides a coherent picture of the phenomenon. The transition is of the saddle-node type, both with and without surfactant. The flow becomes unstable under stationary linear perturbations. Surfactant considerably reduces the interval of stable capillary numbers. Inertia increases the droplet deformation and decreases the critical capillary number. In the presence of the surfactant monolayer, neither the droplet deformation nor the stability is significantly affected by the droplet viscosity. The transient state resulting from instability is fundamentally different for drops with and without surfactant. Tip streaming occurs only in the presence of surfactants. The critical eigenmode leading to tip streaming is qualitatively the same as that yielding the central pinching mode for a clean interface, which indicates that the small local scale characterizing tip streaming is set during the nonlinear droplet deformation. The viscous surface stress does not significantly affect the droplet deformation and the critical capillary number. However, the damping rate of the dominant mode considerably decreases for viscous surfactants. Interestingly, shear viscous surface stress considerably alters the tip streaming arising in the supercritical regime, even for very small surface viscosities. The viscous surface stresses alter the balance of normal interfacial stresses and affect the surfactant transport over the stretched interface.

Spontaneous pulse formation in edgeless photonic crystal resonators

Complex systems are a proving ground for fundamental interactions between components and their collective emergent phenomena. Through intricate design, integrated photonics offers intriguing nonlinear interactions that create new patterns of light. In particular, the canonical Kerr-nonlinear resonator becomes unstable with a sufficiently intense traveling-wave excitation, yielding instead a Turing pattern composed of a few interfering waves. These resonators also support the localized soliton pulse as a separate nonlinear stationary state. Kerr solitons are remarkably versatile for applications, but they cannot emerge from constant excitation. Here, we explore an edge-less photonic-crystal resonator (PhCR) that enables spontaneous formation of a soliton pulse in place of the Turing pattern. We design a PhCR in the regime of single-azimuthal-mode engineering to re-balance Kerr-nonlinear frequency shifts in favor of the soliton state, commensurate with how group-velocity dispersion balances nonlinearity. Our experiments establish PhCR solitons as mode-locked pulses by way of ultraprecise optical-frequency measurements, and we characterize their fundamental properties. Our work shows that sub-wavelength nanophotonic design expands the palette for nonlinear engineering of light.

Bloch oscillations in a Bose–Hubbard chain with single-particle losses

We theoretically investigate Bloch oscillations in a one-dimensional Bose–Hubbard chain, with single-particle losses from the odd lattice sites described by a Lindblad equation. For a single particle the time evolution of the state is completely determined by a non-Hermitian effective Hamiltonian. We analyse the spectral properties of this Hamiltonian for an infinite lattice and link features of the spectrum to observable dynamical effects, such as frequency doubling in breathing modes. We further consider the case of many particles in the mean-field limit leading to complex nonlinear Schrödinger dynamics. Analytic expressions are derived for the generalised nonlinear stationary states and the nonlinear Bloch bands. The interplay of nonlinearity and particle losses leads to peculiar features in the nonlinear Bloch bands, such as the vanishing of solutions and the formation of additional exceptional points. The stability of the stationary states is determined via the Bogoliubov–de Gennes equation and is shown to strongly influence the mean-field dynamics. Remarkably, even far from the mean-field limit, the stability of the nonlinear Bloch bands appears to affect the quantum dynamics. This is demonstrated numerically for a two-particle system.

Dynamics and nonlinear modes of nonlinear saturable PT-symmetric coupler

The existence and stability of nonlinear stationary states in nonlinear saturable coupler with balanced gain–loss (the parity-time symmetric distribution) are studied. Propagation and switching of beams at the different initial excitations are investigated.

Field confinement effect induced by nonlinear interface between nonlinear focusing and defocusing media

We describe the non-symmetrical spatial-distributed electric field near the thin optic layer with nonlinear properties that separated two focusing and defocusing media with Kerr-type nonlinearity differing by values of refractive index. The nonlinear Schrödinger equation (NLSE) with nonlinear potential containing two parameters describes the distribution of electric field strength in adiabatic approximation. The problem is reduced to the solution of NLSEs at the half-spaces with the nonlinear boundary conditions at the interface plane. We obtain five new types of nonlinear stationary states describing the confinement effect of electric field strength across the interface. The field confinement effect is the localization of nonlinear spatially periodic wave during the transition from a one half-space to another one where the field distribution is monotonically damping from the interface. We derive and analyze the frequencies of field confinement effect possibility in dependence of media and interface characteristics.

Critical phenomena and nonlinear dynamics in a spin ensemble strongly coupled to a cavity. I. Semiclassical approach

We present a theoretical study on the nonlinear dynamics and stationary states of an inhomogeneously broadened spin ensemble coupled to a single-mode cavity driven by an external drive with constant amplitude. Assuming a sizeable number of constituents within the ensemble allows us to use a semiclassical approach and to formally reduce the theoretical description to the Maxwell-Bloch equations for the cavity and spin amplitudes. We explore the critical slowing-down effect, quench dynamics, and asymptotic behavior of the system near a steady-state dissipative phase transition accompanied by a bistability effect. Some of our theoretical findings have recently been successfully verified in a specific experimental realization based on a spin ensemble of negatively charged nitrogen-vacancy centers in diamond strongly coupled to a single-mode microwave cavity (see Science Adv. 3, e1701626 (2017)).

The field blocking on the interface with nonlinear response between nonlinear focusing media

We consider the nonlinear nonsymmetrical spatially distributed excitations near the thin layer with nonlinear properties separated with two media with Kerr-type focusing (positive) nonlinearity. We use the nonlinear Schrodinger equation with nonlinear potential containing two parameters to describe the new types of nonlinear stationary states. The problem is reduced to the solution of nonlinear Schrodinger equations on half spaces with the nonlinear boundary conditions at the interface plane. We obtain and analyze the two types of nonlinear excitations describing the effect of the field blocking passing through the interface into one half-space from another one. The field blocking effect is the localization of the nonlinear spatially periodic wave during the transition from one half-space to another one where the wave function is monotonically damping from the interface. We derive the exact energy levels of states of special kinds in explicit form. The conditions of the field blocking effect existence's dependence on the interface and media characteristics are found.

Handling and Stability Analysis of Vehicle Plane Motion

The work analyzes the properties of handling and bicycle vehicle model motion stationary states manifold stability taking into account drift force nonlinear characteristics. Determining single two-axle vehicle nonlinear model stationary states and analyzing their stability were based on a graphical method (Y. M. Pevzner, H. Pacejka). It has its disadvantages: the absence of evident analytical stability criteria for the entire wheeled vehicle circular stationary states manifold. And also the absence of global stability threshold characteristics in the controlled parameter space. The task part suggests developing methods for building bifurcation manifold or critical parameters manifold (longitudinal velocity and wheel turning angle) with which the divergent loss of stability occurs. Known H. Troger, K. Zeman and Fabio Della Rossaa, GiampieroMastinub, Carlo Piccardia results are based on parameter continuation numerical methods which makes the quality analysis of drift force nonlinear characteristics impact on the entire stationary states manifold stability conditions more difficult. A compelling grapho-analytic approach towards bifurcation manifold building and getting circular stationary states analytical stability conditions based on moving from nonlinear drift forces on axles dependencies to their inverse dependence is developed in the suggested work. This methodology allows defining dangerous/safe stability threshold conditions in the control parameters space.

Spatiotemporal wave-train instabilities in nonlinear Schrödinger equation: revisited

A complete description of properties of the wave-train bifurcating from unstable basic oscillatory states (CW nonlinear stationary states) of the nonlinear Schrodinger equation are studied in the moving frames of reference as an initial value problem and using the methods of absolute and convective instabilities. The predictions are in excellent agreement with numerical solutions and may contribute understanding the nonlinear Schrodinger equation complex dynamics under various initial conditions including, localized and/or noisy initial conditions.

Nonlinear Stationary States in PT-Symmetric Lattices

In the present work we examine both the linear and nonlinear properties of two related parity-time (PT)-symmetric systems of the discrete nonlinear Schrödinger (dNLS) type. First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem. Second, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals for a finite PT-dNLS defect in the infinite dNLS lattice are wider than in the case of an isolated PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anticontinuum limit for the dNLS equation. Numerical computations illustrate the existence of nonlinear stationary states as well as the stability and saddle-center bifurcations of discrete solitons.

Loop-structure stability of a double-well-lattice Bose-Einstein condensate

In this work, we consider excited many-body mean-field states of bosons in a double-well optical lattice by investigating stationary Bloch solutions to the non-linear equations of motion. We show that, for any positive interaction strength, a loop structure emerges at the edge of the band structure whose existence is entirely due to interactions. This can be contrasted to the case of a conventional optical (Bravais) lattice where a loop appears only above a critical repulsive interaction strength. Motivated by the possibility of realizing such non-linear Bloch states experimentally, we analyze the collective excitations about these non-linear stationary states and thereby establish conditions for the system's energetic and dynamical stability. We find that there are regimes that are dynamically stable and thus apt to be realized experimentally.

Stationary states of NLS on star graphs

We consider a generalized nonlinear Schrödinger equation (NLS) with a power nonlinearity |ψ|2μψ of focusing type describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the δ potential of strength α on the line, including as a special case (α = 0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive (α < 0, representing a potential well) and repulsive (α > 0, a potential barrier) interaction. In the case of sufficiently strong attractive interaction at the vertex and power nonlinearity μ < 2, including the standard cubic case, we characterize the ground state as minimizer of a constrained action and we discuss its orbital stability. Finally we show that in the free case, for even N only, the stationary states can be used to construct traveling waves on the graph.

Energy exchange, localization, and transfer in nanoscale systems (weak-coupling approximation)

It is shown that both an adequate understanding and effective analysis of energy exchange, localization, and transfer in nanoscale systems is closely related to important aspects of the “wave-particle” duality, one of the key concepts of quantum mechanics. Due to the revealed similarities between the mathematical description of weakly coupled classical oscillators and multilevel quantum systems, this relation turns out to be essential for classical physics as well. Although the development of quantum field theory led, in fact, to the abolition of the wave-particle duality at the fundamental level, putting into the forefront the oscillatory field in the vacuum and imparting to quantized particles the status of field excitations, the wave-particle dilemma continues to be a powerful heuristic principle of theoretical analysis in quantum and classical physics. Within the framework of this approach, it is possible to identify two limiting cases that allow a natural interpretation as applied to polymer physics. In the continuum approximation, very long macromolecules in polymer crystals and other ordered polymer systems (e.g., in the DNA double helix), although considerably more complex in structure than spatially one-dimensional models of nonlinear physics, are ideal objects for the application of ideas and methods of classical nonlinear field theory,. Depending on the conditions of operation, these systems feature both wavelike (normal vibrations and waves) and solitonlike (particlelike) excitation of the nonlinear field, or some combination thereof. The main difficulties to be overcome here are associated with an asymptotic reduction of realistic models of polymer chains and crystals to analytically solvable models. However, for oligomers, with relatively short chains, and nanostructures (second limiting case), it became necessary to develop a fundamentally new approach, since in this case, the formation of localized nonlinear excitations and irreversible energy transfer are preceded (in the level of excitation) by the stage of intense energy exchange between certain groups of particles, referred to as effective particles. Most intense energy transfer is described in terms of limiting phase trajectories, a notion alternative to classical nonlinear normal modes and quantum stationary states. The possibility of introducing effective particles becomes apparent when the initial molecular chain is reduced (in a certain range of its frequencies) to a system of weakly coupled nonlinear oscillators, which is described by the same equations as the multilevel quantum system. With increasing excitation energy, a threshold is reached at which intense energy transfer gives way to the localization of energy on the initially excited effective particle or to its transfer along the chain. In view of the quantum-classical analogy, the analysis of classical linear and nonlinear oscillator models of nanoscale systems is applicable to multilevel quantum systems. In both cases, the asymptotic nature of the wave-particle duality manifests itself, which makes it possible to clarify the relationship between the existing interpretations of it.

Linear and nonlinear behaviors of gyrotron backward wave oscillators

Linear and nonlinear behaviors of gyrotron backward wave oscillators (gyro-BWO) were investigated by both analytical theories and direct numerical calculations. Employing two-scale-length expansion, an analytical linear dispersion relation corresponding to absolute instabilities in a finite-length system has been derived. Detuning from the beam-wave resonance condition due to the finite amplitude radiation fields, meanwhile, was found to play the crucial roles in the nonlinear physics. Near the start oscillation of the gyro-BWO, the radiation field amplitude saturates when the resonance broadening is comparable to the linear growth rate. Far beyond the start oscillation threshold, the beam-wave resonance detuning effectively shortens the interaction length toward that corresponding to the critical oscillation length for the given beam current. The theoretically predicted scaling laws for the linear stability properties and nonlinear stationary states of the gyro-BWO are in good agreement with numerical results.

Oceanic Internal-Wave Field: Theory of Scale-Invariant Spectra

Abstract Steady scale-invariant solutions of a kinetic equation describing the statistics of oceanic internal gravity waves based on wave turbulence theory are investigated. It is shown in the nonrotating scale-invariant limit that the collision integral in the kinetic equation diverges for almost all spectral power-law exponents. These divergences come from resonant interactions with the smallest horizontal wavenumbers and/or the largest horizontal wavenumbers with extreme scale separations. A small domain is identified in which the scale-invariant collision integral converges and numerically find a convergent power-law solution. This numerical solution is close to the Garrett–Munk spectrum. Power-law exponents that potentially permit a balance between the infrared and ultraviolet divergences are investigated. The balanced exponents are generalizations of an exact solution of the scale-invariant kinetic equation, the Pelinovsky–Raevsky spectrum. A small but finite Coriolis parameter representing the effects of rotation is introduced into the kinetic equation to determine solutions over the divergent part of the domain using rigorous asymptotic arguments. This gives rise to the induced diffusion regime. The derivation of the kinetic equation is based on an assumption of weak nonlinearity. Dominance of the nonlocal interactions puts the self-consistency of the kinetic equation at risk. However, these weakly nonlinear stationary states are consistent with much of the observational evidence.

Twin-vortex solitons in nonlocal nonlinear media

We consider soliton formation in thermal nonlinear media bounded by rectangular cross sections and uncover what we believe to be a new class of nonlinear stationary topological state. Specifically, we find that stationary higher-order vortex states in standard shapes do not exist, but rather they take the form of multiple spatially separated single-charge singularities nested in an elliptical beam. Double-charge states are found to be remarkably robust despite their shape asymmetry and phase-singularity splitting. States with higher topological charges are found to be unstable.

Kinetic Description of Nonlinear Plasma Turbulence

The kinetic description of nonlinear plasma turbulence is discussed for Vlasov plasmas with multiple species. Kadomtsev's formalism is reviewed, and the foundation from the Mori method of statistical physics is discussed. The coherent nonlinear term and nonlinear noise term are discussed. In addition to the diagonal term in memory function (eddy damping term), the off-diagonal term in the memory function and nonlinear noise are introduced simultaneously. The off-diagonal term in memory function affects the dynamics as the new nonlinear force, so that a nonlocal interaction in phase space occurs in wave-particle interactions. A set of special solutions for the eddy damping term and off-diagonal coherent term is obtained. The access to a nonlinear stationary state by using an incoherent emission term is explained. Quasi-modes, which are driven by the incoherent emission, are shown to accumulate at a frequency of ω( k )=2ω'( k /2), where ω'( k ) indicates the dispersion relation of the mode. The formulation ...

Launching mechanisms of astrophysical jets

The combination of accretion disks and supersonic jets is used to model many active astrophysical objects, viz., young stars, relativistic stars, and active galactic nuclei. However, existing theories on the physical processes by which these structures transfer angular momentum and energy from disks to jets through viscous or magnetic torques are still relatively approximate. Global stationary solutions do not permit understanding the formation and stability of these structures; and global numerical simulations that include both the disk and jet physics are often limited to relatively short time scales and astrophysically out-of-range values of viscosity and resistivity parameters that are instead crucial to defining the coupling of the inflow/outflow dynamics. Along these lines we discuss self-consistent time-dependent simulations of the launching of supersonic jets by magnetized accretion disks, using high resolution numerical techniques. We shall concentrate on the effects of the disk physical parameters, and discuss under which conditions steady state solutions of the type proposed in the self-similar models of Blandford and Payne can be reached and maintained in a self-consistent nonlinear stationary state.

THE NONLINEAR SCHRÖDINGER EQUATION FOR THE DELTA-COMB POTENTIAL: QUASI-CLASSICAL CHAOS AND BIFURCATIONS OF PERIODIC STATIONARY SOLUTIONS

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight into the features of nonlinear stationary states of periodic potentials. Phenomena well-known from classical chaos are found, such as a bifurcation of periodic stationary states and a transition to spatial chaos. The relation to new features of nonlinear Bloch bands, such as looped and period doubled bands, are analyzed in detail. An analytic expression for the critical nonlinearity for the emergence of looped bands is derived. The results for the delta-comb are generalized to a more realistic potential consisting of a periodic sequence of narrow Gaussian peaks and the dynamical stability of periodic solutions in a Gaussian comb is discussed.