WEAK SOLUTIONS FOR QUASILINEAR BIHARMONIC SYSTEMS

We investigate the entanglement evolution of a two-level atom and a quantized single model electromagnetic filed in the resonance and non-resonance cases. The effects of the initial state, detuning degree, photon number on the entanglement are shown in detail. The results show that the atom-cavity entanglement state appears with periodicity. The increasing of the photon number can make the period of quantum entanglement be shorter. In the non-resonant case, if we choose the suitable initial state the entanglement of atom-cavity can be 1.0

In this paper, we study the quantum coherence dynamics of two-level atom system embedded in non-Markovian reservoir in the presence of classical driving field. We analyze the influence of memory effects, classical driving, and detuning on the quantum coherence. It is found that the quantum coherence has different behaviors in resonant case and non-resonant case. In the resonant case, in stark contrast with previous results, the strength of classical driving plays a negative effect on quantum coherence, while detuning parameter has the opposite effect. However, in non-resonant case through a long time, classical driving and detuning parameter have a different influence on quantum coherence compared with resonant case. Due to the memory effect of environment, in comparison with Markovian regime, quantum coherence presents vibrational variations in non-Markovian regime. In the resonant case, all quantum coherence converges to a fixed maximum value; in the non-resonant case, quantum coherence evolves to different stable values. For zero-coherence initial states, quantum coherence can be generated with evolution time. Our discussions and results should be helpful in manipulating and preserving the quantum coherence in dissipative environment with classical driving field.

In the article a methodology for researching wave processes in weakly nonlinear homogeneous one-dimensional systems – elements of machine and building structures is proposed. It is based on D'Alembert's method of constructing solutions of wave equations and the asymptotic method of nonlinear mechanics. Resonant and non-resonant cases are considered. In the non-resonant case, the influence of viscoelastic forces and small periodic disturbances on the oscillatory process is taken into account. In the resonant case, the influence of the disturbing force and the phase difference of natural oscillations is considered. Numerical methods for linear differential equations were used to analyze the obtained differential equations.

Harmonic solutions for a class of non-autonomous piecewise linear oscillators

The energy transport of the circularly polarized electromagnetic (EM) waves is studied in a collisionless Cairns distributed plasmas, whose constituents are mobile electrons and static ions. For this purpose, the set of Vlasov‒Maxwell equations is employed to derive a modified dispersion relation in terms of plasma dispersion function for parallel propagating EM waves. To study the analytical aspects of the energy transport mechanism and wave‒particle interactions, the plasma dispersion function is expanded for the small and large arguments, obtaining the resonant and non-resonant cases, respectively. It is found that the effects of Cairns parameter, thermal speed, wave frequency and temperature anisotropy significantly influence the curves of the energy flux. One may also note that the exchange of energy of waves with particles is more effective in the resonant case than in the non-resonant case. The outcome of this investigation might be helpful in understanding the wave‒particle interactions in space exploration, and energy deposition in laser-produced plasmas.

A stochastic averaging method is proposed to predict approximately the response of quasi-integrable Hamiltonian systems, i.e., multi-degree-of-freedom integrable Hamiltonian systems subject to lightly linear and (or) nonlinear dampings and weakly external and (or) parametric excitations of Gaussian white noises. According to the present method an n-dimensional averaged Fokker-Planck-Kolmogrov (FPK) equation governing the transition probability density of n action variables or n independent integrals of motion can be constructed in nonresonant case. In a resonant case with α resonant relations, an (n + α)-dimensional averaged FPK equation governing the transition probability density of n action variables and α combinations of phase angles can be obtained. The procedures for obtaining the stationary solutions of the averaged FPK equations for both resonant and nonresonant cases are presented. It is pointed out that the Stratonovich stochastic averaging and the stochastic averaging of energy envelope are two special cases of the present stochastic averaging. Two examples are given to illustrate the application and validity of the proposed method.

It is shown that the structure and property of the exact stationary solution of a stochastically excited and dissipated n-degree-of-freedom Hamiltonian system depend upon the integrability and resonant property of the Hamiltonian system modified by the Wong-Zakai correct terms. For a stochastically excited and dissipated nonintegrable Hamiltonian system, the exact stationary solution is a functional of the Hamiltonian and has the property of equipartition of energy. For a stochastically excited and dissipated integrable Hamiltonian system, the exact stationary solution is a functional of n independent integrals of motion or n action variables of the modified Hamiltonian system in nonresonant case, or a functional of both n action variables and α combinations of phase angles in resonant case with α (1 ≤ α ⩽ n – 1) resonant relations, and has the property that the partition of the energy among n degrees-of-freedom can be adjusted by the magnitudes and distributions of dampings and stochastic excitations. All the exact stationary solutions obtained to date for nonlinear stochastic systems are those for stochastically excited and dissipated nonintegrable Hamiltonian systems, which are further generalized to account for the modification of the Hamiltonian by Wong-Zakai correct terms. Procedures to obtain the exact stationary solutions of stochastically excited and dissipated integrable Hamiltonian systems in both resonant and nonresonant cases are proposed and the conditions for such solutions to exist are deduced. The above procedures and results are further extended to a more general class of systems, which include the stochastically excited and dissipated Hamiltonian systems as special cases. Examples are given to illustrate the applications of the procedures.

The method of successive canonical transformation is developed to study both the classical and quantum versions of a system of n weakly interacting nonlinear oscillators. The method reproduces the essential features of the solutions to these nonlinear differential equations. A general quartic polynomial in the oscillator coordinates is taken as the interaction Hamiltonian. After the canonical transformations to the desired order, the classical system is quantized, resulting in immediate identification of the raising and lowering operators for the perturbed eigenstates in the nonresonant case. In the resonant case (commensurable frequencies) the transformed operators lead to finite-dimensional subspaces within which eigenstates lie, and are obtained by matrix methods. In each case the Heisenberg equations for the operators are solved as fully as presently possible. A particular case of two quantum oscillators is studied in detail, both for the resonant and nonresonant cases. Finally, the method is generalized to the continuum and applied to the Lee model. The usual results for the lowest sector are obtained to second order: the physical V particle and the mass shift of the V. Instability of this particle is related to the occurrence of nonlinear resonance in the lowest-order ``interaction.''

Parametric resonance and bifurcation analysis of thin-walled asymmetric gyroscopic composite shafts: An asymptotic study

We consider quantum analogues of several theorems on Birkhoff normal forms and prove a quantum analogue of the theorem on reducing a Hamiltonian to the normal form and the analogue of the theorem on reducing a Hamiltonian to the real normal form. We obtain the normal form explicitly in the nonresonant case. We consider the uniqueness problems of the normal form and of the normalizing transformation in the nonresonant and resonant cases.

A Dirichelet–Neumann m-point BVP with a p-Laplacian-like operator

The aim of this work is to study the nonlinear temporal evolution of an alternating swirling streaming jet. According to laboratory observations, the viscous dissipation is simulated by means of a phenomenological damping coefficient included in the equation of motion (Jiang et al. in J Fluid Mech 369:273–299, 1998; Wright et al. in J Fluid Mech 402:1–32, 2000) and added to the Bernoulli equation or to the evolution equation. The multiple scales are used for finding out the evolution equations. The fixed points of the solutions have been determined. Really, the modulation concept is exploited in order to analyze the stability criteria in the possible cases of resonance. While in the non-resonant case, the nontrivial solutions are obtained numerically. Different numerical applications have been considered. The studied cases have showed that the Weber number, the phenomenological number and the streamwise circulation play a significant role in determining the dynamics of the developing interfacial patterns.

A perturbation method for the analysis of vibrations and bifurcations associated with non-autonomous systems—I. Non-resonance case

This article investigates nonlocal, quasilinear generalizations of the classical biharmonic operator (-Delta )^2. These fractional p -biharmonic operators appear naturally in the variational characterization of the optimal fractional Poincaré constants in Bessel potential spaces. We study the following basic questions for anisotropic fractional p -biharmonic systems: existence and uniqueness of weak solutions to the associated interior source and exterior value problems, unique continuation properties, monotonicity relations, and inverse problems for the exterior Dirichlet-to-Neumann maps. Furthermore, we show the UCP for the fractional Laplacian in all Bessel potential spaces H^{t,p} for any tin {mathbb R}, 1 le p < infty and s in {mathbb R}_+ {setminus } {mathbb N}: If uin H^{t,p}({mathbb R}^n) satisfies (-Delta )^su=u=0 in a nonempty open set V, then uequiv 0 in {mathbb R}^n. This property of the fractional Laplacian is then used to obtain a UCP for the fractional p -biharmonic systems and plays a central role in the analysis of the associated inverse problems. Our proofs use variational methods and the Caffarelli–Silvestre extension.

Transmission of a Gaussian beam after incidenting nonnormally on a Fabry–Perot etalon: a nonresonant case