arXiv subjects

Stephan I. Tzenov

Publications and source records attributed to Stephan I. Tzenov.

At least 19 recordsLinked to original sources

arXiv2026

The Hamilton-Jacobi Equation and its Application to Nonlinear Beam Dynamics: Comparison of Approaches

The rarely used Hamilton-Jacobi equation has been utilized as an elegant way to find the trajectories of mechanical systems and to derive symplectic maps. Further, the exact solution in kick approximation of Hamilton's equations of motion in interaction representation is written as a generalized one-turn twist map. One can imagine that the nonlinear kick comes first, followed by the one-period rotation along the machine circumference, or a second alternative in which the one-period rotation occurs before the kick. There is a difference in the result of solving Hamilton's equations between the two cases, which is expressed in obtaining a standard forward twist map in the first case, or alternatively a backward map in the second one. This nontrivial and intuitively unclear peculiarity is usually ignored/overlooked in practically all specialized references on the topic. Finally, the statistical properties and the behavior of the density distribution of a particle beam in configuration space under the influence of an isolated sextupole have been studied.

physics.acc-ph↗

arXiv2025

Frobenius-Perron Operator Approach to the Beam-Beam Interaction in Circular Colliders

Unlike most publications devoted to the application of the self-consistent method of the nonlinear Vlasov-Poisson system to the study of beam-beam interaction, in this article an alternative strategy using the elegant approach of the Frobenius-Perron operator for symplectic twist maps has been developed. A detailed analysis of the establishment of an equilibrium density distribution in phase space, as well as the behavior of the perturbed distribution function with respect to the coherent stability of the two beams, has been carried out. Using the Renormalization Group technique for the reduction of the Frobenius-Perron operator, the case where the unperturbed rotation frequency (unperturbed betatron tune) of the map is far from any structural resonance driven by the beam-beam kick perturbation has been analyzed in detail. It has been shown that up to second order in the beam-beam parameter, the renormalized map propagator with nonlinear stabilization describes a random walk of the angle variable, implying that there exists an equilibrium distribution function depending only on the action variable. The linearized Frobenius-Perron operators for each beam imply a discrete form of the linearized Vlasov equations, which essentially comprises a new method for calculating coherent beam-beam instabilities using a matrix mapping technique. In the special case of an isolated coherent beam-beam resonance, a stability criterion for coherent beam-beam resonances has been found in closed form. An intriguing particular concerning the effect of repeated beam-beam collisions on collider luminosity has been derived explicitly. An addition of luminosity per kick (small though, of the order of the beam-beam parameter) in the course of successive beam-beam collisions could be achieved.

physics.acc-ph↗

arXiv2024

Nonlinear Cnoidal Waves and Formation of Patterns and Coherent Structures in Intense Charged Particle Beams

The longitudinal dynamics of an intense high energy beam moving in a resonator cavity has been studied in some detail. Through the method of separation of variables and its obvious straightforward generalization, a solution of the Vlasov equation for the distribution function of an intense charged particle beam in the longitudinal direction has been obtained. The thus found Bernstein-Greene-Kruskal (BGK) equilibrium has been utilized to construct stationary wave patterns in the special case when the velocity distribution (energy error distribution) is Maxwellian. These are cnoidal wave patterns, showing rather intriguing and in a sense unexpected analogy between the equilibrium wave patterns in an intense charged particle beam and similar wave clusters originally observed in shallow water. Based on the hydrodynamic model, fully equivalent to the coupled nonlinear system of the Vlasov equation for the distribution function of an intense beam in the longitudinal direction and the equation for the resonator cavity potential, an amplitude equation in the most general form has been derived. A very interesting and important property of the nonlinear amplitude equation is the fact that it is of hyperbolic type (nonlinear wave equation with complex coefficients) in the entire interval of admissible values for the wave number except for a single critical point, in which it is of parabolic type (non-linear Schrodinger equation with complex coefficients).

physics.plasm-ph↗

arXiv2024

The Method of Formal Series: Applications to Nonlinear Beam Dynamics and Invariants of Motion

A novel technique to determine invariant curves in nonlinear beam dynamics based on the method of formal series has been developed. It is first shown how the solution of the Hamilton equations of motion describing nonlinear betatron oscillations in the presence of a single sextupole can be represented in a nonperturbative form. Further, the solution of the Hamilton-Jacobi equation is obtained in a closed symbolic form as a ratio of two series in the perturbation parameter (and the nonlinear action invariant), rather than a conventional power series according to canonical perturbation theory. It is well behaved even for large values of the perturbation parameter close to strong structural resonances. The relationship between existing invariant curves and the so-called scattering orbits in classical scattering theory has been revealed.

physics.acc-ph↗

arXiv2023

Linear Coupling of Transverse Betatron Oscillations. Dynamic Stability and Invariants of Motion

Based on the technique of the discrete one-turn transfer maps, the problem of linear coupling between horizontal and vertical betatron oscillations in an accelerator has been treated exactly and entirely in explicit form. The stability region in the fractional part of the horizontal and the vertical betatron tune space as a function of the linear coupling strength, has been obtained, and the increment/decrement of the horizontal and the vertical betatron oscillations in the case of the linear sum resonance has been shown to be approximately equal to the half of the coupling strength. The normal form parameterization of the one-turn linear map with horizontal-to-vertical coupling has been developed in detail in the spirit of the Edwards and Teng formalism. The motion in the normal mode in the new normal form coordinates is decoupled implying that two independent Courant-Snyder invariants exist, which have been found explicitly.

physics.acc-ph↗

arXiv2023

Quantum Description of Free Electron Laser Radiation and Nonlinear Amplitude Equations

A relativistic quantum mechanical model to describe the quantum FEL dynamics has been developed. Neglecting the spin of electrons in the impacting beam, this model is based on the Klein-Gordon equation coupled to the Poisson equation for the space-charge potential and the wave equation for the transverse components of the radiation field. Furthermore, a system of coupled nonlinear envelope equations for the slowly varying amplitudes of the electron beam distribution and the radiation field has been derived. The fundamental system of basic equations have been cast into a suitable hydrodynamic formulation. In the framework of the hydrodynamic representation, a new dispersion relation has been derived and analyzed in both the quantum and the quasi-classical regimes, where the space-charge oscillations of the electron beam are taken into account.

physics.plasm-ph↗

arXiv2019

Dispersion Properties, Nonlinear Waves and Birefringence in Classical Nonlinear Electrodynamics

Using the very basic physics principles, we have studied the implications of quantum corrections to classical electrodynamics and the propagation of electromagnetic waves and pulses. The initial nonlinear wave equation for the electromagnetic vector potential is solved perturbatively about the known exact plane wave solution in both the free vacuum case, as well as when a constant magnetic field is applied. A nonlinear wave equation with nonzero convective part for the (relatively) slowly varying amplitude of the first-order perturbation has been derived. This equation governs the propagation of electromagnetic waves with a reduced speed of light, where the reduction is roughly proportional to the intensity of the initial pumping plane wave. A system of coupled nonlinear wave equations for the two slowly varying amplitudes of the first-order perturbation, which describe the two polarization states, has been obtained for the case of constant magnetic field background. Further, the slowly varying wave amplitude behavior is shown to be similar to that of a cnoidal wave, known to describe surface gravity waves in shallow water. It has been demonstrated that the two wave modes describing the two polarization states are independent, and they propagate at different wave frequencies. This effect is usually called nonlinear birefringence.

physics.plasm-ph↗

arXiv2018

Relativistic Nonlinear Whistler Waves in Cold Magnetized Plasmas

Starting from the Vlasov-Maxwell equations describing the dynamics of various species in a quasi-neutral plasma immersed in an external solenoidal magnetic field and utilizing a technique known as the hydrodynamic substitution, a relativistic hydrodynamic system of equations governing the dynamics of various species has been obtained. Based on the method of multiple scales, a system comprising three nonlinear Schrodinger equation for the transverse envelopes of the three basic whistler modes, has been derived. Using the method of formal series of Dubois-Violette, a traveling wave solution of the derived set of coupled nonlinear Schrodinger equations in both the relativistic and the non relativistic case has been obtained. An intriguing feature of our description is that whistler waves do not perturb the initial uniform density of plasma electrons. The plasma response to the induced whistler waves consists in transverse velocity redistribution, which follows exactly the behaviour of the electromagnetic whistlers. This property may have an important application for transverse focusing of charged particle beams in future laser plasma accelerators. Yet another interesting peculiarity are the selection rules governing the nonlinear mode coupling. According to these rules self coupling between modes in the non relativistic regime is absent, which is a direct consequence of the vector character of the interaction governed by the Lorentz force.

physics.plasm-ph↗

arXiv2017

Nonlinear Waves and Coherent Structures in Quasi-neutral Plasmas Excited by External Electromagnetic Radiation

Starting from the Vlasov-Maxwell equations describing the dynamics of various species in a quasi-neutral plasma, an exact relativistic hydrodynamic closure for a special type of water-bag distributions satisfying the Vlasov equation has been derived. It has been shown that the set of equations for the macroscopic hydrodynamic variables coupled to the wave equations for the self-consistent electromagnetic field is fully equivalent to the Vlasov-Maxwell system. Based on the method of multiple scales, a system comprising a vector nonlinear Schrodinger equation for the transverse envelopes of the self-consistent plasma wakefield, coupled to a scalar nonlinear Schrodinger equation for the electron current velocity envelope, has been derived. Using the method of formal series of Dubois-Violette, a traveling wave solution of the derived set of coupled nonlinear Schrodinger equations in the case of circular wave polarization has been obtained. This solution is represented as a ratio of two formal Volterra series. The terms of these series can be calculated explicitly to every desired order.

physics.plasm-ph↗

arXiv2014

Strange Attractors Characterizing the Osmotic Instability

In the present paper a simple dynamical model for computing the osmotically driven fluid flow in a variety of complex, non equilibrium situations is derived from first principles. Using the Oberbeck-Boussinesq approximation, the basic equations describing the process of forward osmosis have been obtained. It has been shown that these equations are very similar to the ones used to model the free Rayleigh-Benard convection. The difference is that while in the case of thermal convection the volume expansion is driven by the coefficient of thermal expansion, the key role for the osmotic instability is played by the coefficient of isothermal compressibility. In addition, it has been shown that the osmotic process represents a propagation of standing waves with time-dependent amplitudes and phase velocity, which equals the current velocity of the solvent passing through the semi-permeable membrane. The evolution of the amplitudes of the osmotic waves is exactly following the dynamics of a strange attractor of Lorenz type with corresponding control parameters.

physics.flu-dyn↗

arXiv2011

Hydrodynamic Approach to the Free Electron Laser Instability

Starting from the Vlasov-Maxwell system, an exact relativistic hydrodynamic closure for a special type water bag distributions satisfying the Vlasov equation has been derived. It has been shown that the hydrodynamic equations are fully equivalent to the original Vlasov-Maxwell equations. The linear stability of the exact hydrodynamic closure has been studied as well. It has been shown that all basic features of the small signal gain can be derived from the fluid dynamic description developed here. Satisfactory agreement with previously reported results has been also found.

physics.acc-ph↗

arXiv2011

Nonlinear Waves and Coherent Structures in the Quantum Single-Wave Model

Starting from the von Neumann-Maxwell equations for the Wigner quasi-probability distribution and for the self-consistent electric field, the quantum analog of the classical single-wave model has been derived. The linear stability of the quantum single-wave model has been studied, and periodic in time patterns have been found both analytically and numerically. In addition, some features of quantum chaos have been detected in the unstable region in parameter space. Further, a class of standing-wave solutions of the quantum single-wave model has also been found, which have been observed to behave as stable solitary-wave structures. The analytical results have been finally compared to the exact system dynamics obtained by solving the corresponding equations in Schrodinger representation numerically.

physics.plasm-ph↗

arXiv2011

Nonlinear Density Waves in the Single-Wave Model

The single-wave model equations are transformed to an exact hydrodynamic closure by using a class of solutions to the Vlasov equation corresponding to the waterbag model. The warm fluid dynamic equations are then manipulated by means of the renormalization group method. As a result, amplitude equations for the slowly varying wave amplitudes are derived. Since the characteristic equation for waves has in general three roots, two cases are examined. If all three roots of the characteristic equation are real, the amplitude equations for the eigenmodes represent a system of three coupled nonlinear equations. In the case, where the dispersion equation possesses one real and two complex conjugate roots, the amplitude equations take the form of two coupled equations with complex coefficients. The analytical results are then compared to the exact system dynamics obtained by solving the hydrodynamic equations numerically.

physics.plasm-ph↗

arXiv2009

Dynamics of Particles in Non Scaling FFAG Accelerators

Non scaling Fixed-Field Alternating Gradient (FFAG) accelerators have an unprecedented potential for muon acceleration, as well as for medical purposes based on carbon and proton hadron therapy. They also represent a possible active element for an Accelerator Driven Subcritical Reactor (ADSR). Starting from first principle the Hamiltonian formalism for the description of the dynamics of particles in non scaling FFAG machines has been developed. The stationary reference (closed) orbit has been found within the Hamiltonian framework. The dependence of the path length on the energy deviation has been described in terms of higher order dispersion functions. The latter have been used subsequently to specify the longitudinal part of the Hamiltonian. It has been shown that higher order phase slip coefficients should be taken into account to adequately describe the acceleration in non scaling FFAG accelerators. A complete theory of the fast (serpentine) acceleration in non scaling FFAGs has been developed. An example of the theory is presented for the parameters of the Electron Machine with Many Applications (EMMA), a prototype electron non scaling FFAG to be hosted at Daresbury Laboratory.

physics.acc-ph↗

arXiv2006

Stochastic Properties of the Frobenius-Perron Operator

In the present paper the Renormalization Group (RG) method is adopted as a tool for a constructive analysis of the properties of the Frobenius-Perron Operator. The renormalization group reduction of a generic symplectic map in the case, where the unperturbed rotation frequency of the map is far from structural resonances driven by the kick perturbation has been performed in detail. It is further shown that if the unperturbed rotation frequency is close to a resonance, the reduced RG map of the Frobenius-Perron operator (or phase-space density propagator) is equivalent to a discrete Fokker-Planck equation for the renormalized distribution function. The RG method has been also applied to study the stochastic properties of the standard Chirikov-Taylor map.

nlin.CD↗

arXiv2005

Irrotational Momentum Fluctuations Conditioning the Quantum Nature of Physical Processes

Starting from a simple classical framework and employing some stochastic concepts, the basic ingredients of the quantum formalism are recovered. It has been shown that the traditional axiomatic structure of quantum mechanics can be rebuilt, so that the quantum mechanical framework resembles to a large extent that of the classical statistical mechanics. The main assumption used here is the existence of a random irrotational component in the classical momentum. Various basic elements of the quantum formalism (calculation of expectation values, the Heisenberg uncertainty principle, the correspondence principle) are recovered by applying traditional techniques, borrowed from classical statistical mechanics.

quant-ph↗

arXiv2005

Generation and Propagation of Nonlinear Waves in Travelling Wave Tubes

The generation and evolution of nonlinear waves in microwave amplifiers such as travelling wave tubes, free electron lasers and klystrons have been studied. The analysis is based on the hydrodynamic and field equations for the self-consistent evolution of the beam density distribution, the current velocity and the electromagnetic fields. A system of coupled nonlinear Schrödinger equations for the slowly varying amplitudes of interacting beam-density waves has been derived. Under the approximation of an isolated mode neglecting the effect of the rest of the modes, this system reduces to a single nonlinear Schrödinger equation for that particular mode.

physics.acc-ph↗

arXiv2004

Whistleron Gas in Magnetized Plasmas

We have studied the nonlinear dynamics of whistler waves in magnetized plasmas. Since plasmas and beam-plasma systems considered here are assumed to be weakly collisional, the point of reference for the analysis performed in the present paper is the system of hydrodynamic and field equations. We have applied the renormalization group method to obtain dynamical equations for the slowly varying amplitudes of whistler waves. Further, it has been shown that the amplitudes of eigenmodes satisfy an infinite system of coupled nonlinear Schrödinger equations. In this sense, the whistler eigenmodes form a sort of a gas of interacting quasiparticles, while the slowly varying amplitudes can be considered as dynamical variables heralding the relevant information about the system. An important feature of our description is that whistler waves do not perturb the initial uniform density of plasma electrons. The plasma response to the induced whistler waves consists in velocity redistribution which follows exactly the behaviour of the whistlers. In addition, selection rules governing the nonlinear mode coupling have been derived, which represent another interesting peculiarity of our description.

physics.plasm-ph↗