Parametric Excitation of Plasma instabilities by two Monochromatic Pump waves

The dispersion characteristics of a plasma in a pump field ??(t) = ?? sub ω₀t + ??₁ sin ω₁t are considered. Firstly we assume, that the second wave is weak (|??₁| ? |??₀|) and the frequency ω₁ is near sω₀(ω₁ = sω₀ + Ω,Ω ? ω₀). We obtain the dispersion equation, describing the parametric coupling of the waves driven by the strong field ??₀ sin ω₀t under the resonance condition ω₀ ≈ ω_Le/P and derive the expressions for the growth rates (ω_Le is the electron LANGMUIR frequency; s, p are integers). In the second part it is shown, that a strong field ??₁ with a frequency ω₁ much larger than ω _Le(ω _Le ≈ pω₀) stabilizes the plasma; the growth rates are reduced and the frequency region of the parametric instability is contracted.     

[Russian Text Ignored]

The parametric excitation of longitudinal waves in an infinite homogeneous plasma by a pump field E (t) = E ₀(t) sin (ω₀t + φ(t)) is studied on the basis of the Vlasov equation, where the amplitude E ₀(t) and the phase φ(t) are slowly varying compared with ω₀ periodic functions. Firstly it is assumed that ω₀ is much larger than the electron plasma frequency ω_Le. In the second part the parametric instabilities are considered under the resonance condition ω₀ ≈ ω_Le. In both parts the threshold fields for the excitation of the longitudinal waves and their growth rates are calculated. As an example these values are analysed for both a sequence of pump impulses and a phase-moduated pump field. They are compared with the results received for a monochromatic pump field.     

Dynamical Correlation Functions for an Impenetrable Bose Gas with Neumann or Dirichlet Boundary Conditions

Abstract

We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions ψ(x ₁, 0)ψ ^?(x ₂ , t) _±,T . We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case x ₁ = 0, we express correlation functions with Neumann boundary conditions ψ(0, 0)ψ ^?(x ₂ , t) _+,T , in terms of solutions of nonlinear partial differential equations which were introduced in [1] as a generalization of the nonlinear Schrödinger equations. We generalize the Fredholm minor determinant formulae of ground state correlation functions ψ(x ₁)ψ ^?(x ₂) _±,0 in [2], to the Fredholm determinant formulae for the time and temperature dependent correlation functions ψ(x ₁, 0)ψ ^?(x ₂ , t) _±,T , t ∈ R, T ≥ 0.     

Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions

If X = X(t, ξ) is the solution to the stochastic porous media equation in O ì R^d, 1 ￡ d ￡ 3,{\mathcal{O}\subset \mathbf{R}^d, 1\le d\le 3,} modelling the self-organized criticality (Barbu et al. in Commun Math Phys 285:901–923, 2009) and X _c is the critical state, then it is proved that ò^￥₀m(O\O^t₀)dt < ￥,\mathbbP-a.s.{\int^{\infty}_0m(\mathcal{O}{\setminus}\mathcal{O}^t_0)dt<{\infty},\mathbb{P}\hbox{-a.s.}} and lim_t?￥ ò_O|X(t)-X_c|dx = l < ￥, \mathbbP-a.s.{\lim_{t\to{\infty}} \int_\mathcal{O}|X(t)-X_c|d\xi=\ell<{\infty},\ \mathbb{P}\hbox{-a.s.}} Here, m is the Lebesgue measure and O^t_c{\mathcal{O}^t_c} is the critical region {x ? O; X(t,x)=X_c(x)}{\{\xi\in\mathcal{O}; X(t,\xi)=X_c(\xi)\}} and X _c (ξ) ≤ X(0, ξ) a.e. x ? O{\xi\in\mathcal{O}}. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), lim_{t ? ￥} ò_K|X(t)-X_c|dx = 0{\lim_{t \to {\infty}} \int_K|X(t)-X_c|d\xi=0} exponentially fast for all compact K ì O{K\subset\mathcal{O}} with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case ℓ = 0.     

On the oscillons in the signum-Gordon model

New periodic solutions of signum-Gordon equation are presented. We first find solutions φ₀(x, t) defined for (x, t) ∈ ? × [0, T ] and satisfying the condition φ₀(x, 0) = φ₀(x, T ) = 0. Then these solutions are extended to the whole spacetime by using (2.4).     

The modified Korteweg-de Vries equation on the quarter plane with t-periodic data

We study the modified Korteweg-de Vries equation posed on the quarter plane with asymptotically t-periodic Dirichlet boundary datum u(0,t) in the sense that u(0,t) tends to a periodic function g?₀ (t) with period τ as t → ∞. We consider the perturbative expansion of the solution in a small ε > 0. Here we show that if the unknown boundary data u_x(0,t) and u_xx(0,t) are asymptotically t-periodic with period τ which tend to the functions g?₁ (t) and g?₂ (t) as t → ∞, respectively, then the periodic functions g?₁ (t) and g?₂ (t) can be uniquely determined in terms of the function g?₀ (t). Furthermore, we characterize the Fourier coefficients of g?₁ (t) and g?₂ (t) to all orders in the perturbative expansion by solving an infinite system of algebraic equations. As an illustrative example, we consider the case of a sine-wave as Dirichlet datum and we explicitly determine the coefficients for large t up to the third order in the perturbative expansion.     

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

Abstract

Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u _t = (k(u) u _x)_x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)).     

Arbitrarily slow decay of correlations in quasiperiodic systems

For diffusive motion in random media it is widely believed that the velocity autocorrelation functionc(t) exhibits power law decay as time;t. We demonstrate that the decay ofc(t) in quasiperiodic media can be arbitrarily slow within the class of integrable functions. For example, ind=1 with a potentialV(x)=cosx+coskx, there is a dense set of irrationalk's such that the decay ofc(k, t) is slower than 1/t ⁽¹⁺⁾ for any>0. The irrationals producing such a slow decay ofc(k, t) arevery well approximated by rationals.     

Evolution of the density for a chaotic map

For the discrete-time map x_t+11 = 4x_t(1?x_t) an exact, explicit expression is given for the time-dependent density r_t (x) evolving from a uniform initial density on (0,1). As t → ∞, r_t(x) approaches the known invariant density $\text{r(x) = 1/[π}\text{x(1?x)}\text{]}$.     

Ionization of Coulomb Systems in {\mathbb{R}^3} by Time Periodic Forcings of Arbitrary Size

We analyze the long time behavior of solutions of the Schrödinger equation ${i\psi_t=(-\Delta-b/r+V(t,x))\psi}We analyze the long time behavior of solutions of the Schr?dinger equation iy_t=(-D-b/r+V(t,x))y{i\psi_t=(-\Delta-b/r+V(t,x))\psi}, x ? \mathbbR³{x\in\mathbb{R}^3}, r =  |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) =  V(t +  2π/ω, x) with zero time average.     

Elastic Scattering of Electrons by a Cut-Off Coulomb Potential at Low Energies

Elastic scattering of electrons by cut-off Coulomb potential U_c(r) is investigated, where U_c(r) = 0, for r > r_c and U_c(r) = ?1/r + 1/r_c for r ≦ r_c. This is first considered in terms of classical, and later quantum mechanical (partial wave) methods in the low energy range 0 ≦ ? ? 1/r_c, where ? is the energy of the free electron. Scattering in this energy region displays a number of particular characteristics, such as back scattering, at certain energies. It can be concluded that some agreement does exist between the classical and quantum mechanical results.     

The correlation function for a quantum oscillator in a low-temperature heat bath

The momentum autocorrelation function c(t) for a quantum oscillator coupled with harmonic forces to a heat bath of oscillators is calculated at low temperatures. It is found that c(t) contains two distinct terms: one, the zero-point contribution c₀(t), is temperature independent, and the other, c₁(t), does depend on temperature. We concentrate our attention on the low-temperature case. An expression for c₁(t) is obtained, which is valid for arbitrary strenghts of the coupling and for arbitrary times. It is shown that c₁(t) is governed by the low-frequency behaviour of F(λ) = A²(λ)ϱ(λ), where ϱ(λ) is the density of normal modes and A(λ) is the central-oscillator component of the λth normal mode; other details of the problem are irrelevant. It is found that c₁(t) decays in time as an inverse-power law, with a relaxation time t_q ≈ ħ/kT.     

Stability for the Korteweg-de Vries equation by inverse scattering theory

The stability problems for the Korteweg-de Vries equation, where linear stability fails, are investigated by the inverse scattering method. A rather general solution u(t, x) of the K-dV equation is shown to depend, for fixed time t, continuously on the initial condition u(0, x). For a continuum solution u_c(t, x), this continuity holds uniformly in t (stability), but for a soliton solution this is not true. A soliton solution can be uniquely decomposed into a continuum and discrete (soliton) part: u(t, x) = u_e(t, x) + u_d(t, x). Then the perturbed solution u is close to u after a suitable t-dependent “shift” of the soliton part (form stability).     

Coexistence of superconductivity and magnetic order in YBa2Cu4O8

⁵⁷Fe Mössbauer spectroscopy, magnetic susceptibility and powder x-ray-diffraction measurements were used to study superconductivity and magnetic order in YBa₂(Cu_1?xFe_x)₄O_8+δ. T_c is decreasing with x, disappearing for x>x_c≈0.04. For xc iron substitutes Cu, predominantly in the Cu(1) site exhibiting a single quadrupole Mössbauer spectrum at 90 K. For x>x_c magnetic order is observed in the Cu(2) site, T_N=380 (5) K for x=0.1 and H_eff (Cu(2), 4.2 K)=510(2) kOe. However, the most surprising discovery is that for x=0.025, for which T_c=27(2) K, the Fe in the Cu(1) site orders magnetically at T_N=30(2) K and H_eff (Cu(1), 4.2 K)=461(2) kOe. The coexistence and competition between superconductivity and magnetic order in the Cu(1) and Cu(2) sites in YBa₂Cu₄O₈ are discussed in terms of the previously observed phase diagrams for Y_1?xPr_xBa₂(Cu_1?yFe_y)₃O_z.     

Anomalous behavior of the coercive force of dilute nickel ferrites NiGaxAlxFe2 ? 2 xO4 with a frustrated magnetic structure

The behavior of the spontaneous magnetization σ _s (T) and coercive force H _c (T) of dilute nickel ferrites NiGa _x Al _x Fe_{2 − 2x} O₄ (x = 0, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, and 0.8) has been studied. Above the transition temperature T _t , the coercive force H _c is found to reveal an anomalous behavior for compositions with x ≥ 0.4, namely, the temperature dependence of the coercive force H _c (T) exhibits a maximum in the range from T _t to T _C. For the reduced temperature θ₂ = 0.8 T _C, at which ferrites with the substitution x ≥ 0.4 reside in the spin glass state, the coercive force H _c is observed to increase sharply with x. The assumption is made that the clusters prevailing in the spin glass state are no larger than 3 nm in size. Original Russian Text ? L.G. Antoshina, A.B. Korshak, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 5, pp. 900–903.     

Nonlinear susceptibility x;(3) in a direct gap semiconductor derived from coupled inter-and intraband equations of motion

A set of generalized Maxwell Bloch Equations describing a two band semiconductor as a system of interacting two level atoms is used to calculate the nonlinear dielectric susceptibility x;⁽³⁾ (ω, ? ω, ω). For near resonance conditions (ω ≈ ω_g) we find a simplified expression for χ⁽³⁾ that is compared with other theories. A fit using parameters relevant for InSb leads to good agreement with experimental results.     

Phonons,Diffusons, and the Boson Peak in Two-Dimensional Lattices with Random Bonds

Within the model of stable random matrices possessing translational invariance, a two-dimensional (on a square lattice) disordered oscillatory system with random strongly fluctuating bonds is considered. By a numerical analysis of the dynamic structure factor S(q, ω), it is shown that vibrations with frequencies below the Ioffe-Regel frequency ω_IR are ordinary phonons with a linear dispersion law ω(q) ∝ q and a reciprocal lifetime б ~ q³. Vibrations with frequencies above ω_IR, although being delocalized, cannot be described by plane waves with a definite dispersion law ω(q). They are characterized by a diffusion structure factor with a reciprocal lifetime б ~ q², which is typical of a diffusion process. In the literature, they are often referred to as diffusons. It is shown that, as in the three-dimensional model, the boson peak at the frequency ωb in the reduced density of vibrational states g(ω)/ω is on the order of the frequency ω_IR. It is located in the transition region between phonons and diffusons and is proportional to the Young’s modulus of the lattice, ω _b ≃E.

     

A simple and direct periodic solution of the Korteweg-de vries equations

The solutionq(x, t) of one of the KdV hierarchy is assumed to be a potential in the Schrödinger equation as usual. We differentiate this equation with respect to the time variable and solve it with the aid of the Green function. The obtained equation relatesw _t (x, t, λ)=φ _t (x + c, x, t, λ) withq _t (x, t). The functionφ(x, x ₀,t, λ) obeys the Schrödinger equation and the boundary conditionsφ(x ₀,x ₀,t, λ)=0,φ _x (x ₀,x ₀;t, λ)=1. The shiftingc is equal to the period. We differentiatew _t (x, t, λ) three times with respect to thex coordinate and obtain the time derivative of the Milne equation. The integration of this equation with respect tox allows to solve simply the inverse problem. The reconstructed periodic potential is given by means of the well known formula for the root functions ofw(x, t, λ). The time behaviour of this function, i.e. the solution of the KdV equation, is obtained when one replacesq _t (x, t) by an expression of the KdV hiearchy in the relation betweenq _t (x, t) andw _t (x, t, λ) and transforms it. We estimated also the limit, whenc → ∞, i.e. the possible relation of the periodic solutions with the soliton ones.     

On the asymptotic behaviour of the mass operator near the Fermi energy

By taking due account of momentum conservation, it is shown that, when ω is near the Fermi energy ω_F, the imaginary part of the mass operator $\text{M(}\text{k}\text{, ω)}$ for an infinite Fermi system behaves like (ω ? ω_F)^p(k) where the exponent p(k) ? 2 depends on the interval in which $\text{|}\text{k}\text{|}$ is lying. In particular, the commonly asserted quadratic behaviour (ω ? ω_F² is shown to be true only for $\text{|}\text{k}\text{| ? 3k}{}_{\mathrm{<mtext>F</mtext>}}$. It is explicity assumed that the Fermi system admits a perturbative type treatment.     

N NMR Echo in NH4ClO4after the Pulse Sequence (θ1)x–τ–(θ2)y–t

We have derived a closed-form expression for the solid echo signal of quadrupolarI= 1 nuclei after the pulse sequence (θ₁)_x–τ–(θ₂)_y–tfor arbitrary values of the RF nutation frequency ω₁= γB₁and the quadrupolar frequency ω_Q. In the case of single crystals both the true echo term of this expression and its induction-signal-like terms are important as shown by experiments on¹⁴N nuclei in NH₄ClO₄crystal. Conditions for obtaining the maximal echo in powder samples are presented. A very lowB₁field together with long RF pulses may distort even the central part of the spectrum, resulting in strange looking apparent spectra.