Volume 45, Issue 7

In 1925 Cherry [1] discussed two oscillators of positive and negative energy that are nonlinearly coupled in a special way, and presented a class of exact solutions of the nonlinear equations showing explosive instability independent of the strength of the nonlinearity and the initial amplitudes. In this paper Cherry's Hamiltonian is transformed into a form which allows a simple physical interpretation. The new Hamiltonian is generalized to three nonlinearly coupled oscillators; it corresponds to three-wave interaction in a continuum theory, like the Vlasov-Maxwell theory, if there exist linear negative energy waves [2-4, 5, 6], Cherry was able to present a two-parameter solution set for his example which would, however, allow a four-parameter solution set, and, as a first result, an analogous three-parameter solution set for the resonant three-oscillator case is obtained here which, however, would allow a six-parameter solution set. Nonlinear instability is therefore proven so far only for a very small part of the phase space of the oscillators. This paper gives in addition the complete solution for the three-oscillator case and shows that, except for a singular case, all initial conditions, especially those with arbitrarily small amplitudes, lead to explosive behaviour. This is true of the resonant case. The non-resonant oscillators can sometimes also become explosively unstable, but the initial amplitudes must not be infinitesimally small. A few examples are presented for illustration.

Onsager's result concerning the quantization of the magnetic flux enclosed by the orbit of an electron moving in a static uniform magnetic field is shown to hold also in the relativistic case, and not to be restricted to the regime considered by Onsager in which the correspondence principle can be applied. The results obtained via the correspondence principle and via expectation values differ, however, in the deails. It is shown also that the eigenenergies and eigenfunctions of Dirac's relativistic equation for an electron in a static non-uniform and arbitrarily strong magnetic field may be obtained directly from the solutions of Schrödinger's non-relativistic equation. In this case the expectation value of an observable depending on coordinates only equals the one calculated directly with the non-relativistic eigenfunctions.

NMR measurements of the 115 In-Knightshift and the linewidth in liquid In-Mn are reported. They were carried out in an external magnetic field of about 4.0 T at concentrations 0, 3, and 6 at.% Mn at temperatures from 400 K to 1300 K. According to our experimental results there are no localized magnetic moments in the liquid In-Mn-system. The experimental results are discussed in the framework of theories on metallic alloys.

We derive an effective interaction between hydrogen impurities in Niobium using a microscopic theory of hydrogen in metals. Our model consists of an infinite bcc-crystal with two hydrogen centres occupying tetrahedral interstitial sites, neighbouring on the first to the fourth coordination shell. The elastic interactions are assumed to obey the classical harmonic approximation. The electronic interactions due to both the coulomb potentials and the overlap of the impurity induced electron densities in the vicinity of the interstitials also play an important role. These latter interactions are treated as two-body interactions in a zeroth order approximation of the New Tamm-Dancoffmethod. A separation ansatz results in an effective interaction which depends on the distance between the interstitials and upon the spin states of their excess electrons. We propose some improvements on our model, and to test our calculations, we construct the grand partition function of an appropriate lattice gas model.

Based on continuum theory, general expressions are derived which indicate how the electric field induced stripe instability threshold of twisted nematic layers is influenced by the liquid crystal material parameters. Results are applied to supertwist displays

Gauss's error law for generalized jump processes in which the average value is also the most probable value is derived from the maximum likelihood path for the growth of a fluctuation. Evolution towards equilibrium and the spontaneous growth of a fluctuation from equilibrium are analyzed and compared in the continuous, diffusion limit.

According to the second law of thermodynamics, thermal interactions of material bodies lead to an increase in entropy. Black-hole thermodynamics has unknowingly repudiated this law of Nature.

Two sterically hindered derivatives of dimethylaminobenzonitrile (DMCA) have been synthesized and the dipole moment of their fluorescent excited state has been determined in differently polar solvents. Its value increases with solvent polarity up to a limiting value of (50 + 2.5) • 10 -30 Cm. The results are discussed with reference to the twisted intramolecular charge transfer (TICT) state formation in the class of molecules related to DMCA.

The multipoint Wronskian method is applied to one-dimensional vibrational eigenvalue problems with two different potential functions. Systematic ways of selecting points are discussed. The errors introduced by interpolation methods and the effect of including higher derivatives of the potential are analyzed.

Lattice dynamical calculations of the spinel type MCr 2 S 4 (M = Mn, Fe, Cd) have been done using various potential models (short-range, rigid-ion, polarizable-ion). The main results are that (i) the vibrational modes (eigenvectors) and potential energy distributions of the Raman and IR allowed phonon modes of the three chromium sulfides are very similar, (ii) the A-X and B-X short-range force constants (referring to AB2X4) strongly depend on the structural parameter u, i.e., the tetrahedral A-X force constants are smaller than the respective octahedral B-X ones opposite to previous calculations on the basis of an ideal spinel structure with u = 0.25, (iii) bending force constants (X-A-X and X-B-X), but not X-X and B-B repulsive forces, are negligible, (iv) in the case of the breathing mode of the tetrahedral AX 4 unit (species A 1g ) the demand on B-X and X-X (stretching and repulsion) forces is larger than that on the A-X force, and (v) the effective dynamic charges of the bivalent metal ions are nearly zero.

The limit cycle behaviour of the intensity and of the polarization of a monochromatic two-polarization- mode laser with an intensity coupling asymmetry and a linear phase anisotropy is studied analytically. In a previous paper we found such limit cycle oscillations numerically. Explicit formulae are derived by perturbation methods for the case of small intensity coupling asymmetry and for the case of large pumping, respectively, which describe the dependence on the laser parameters of the limit cycle period as well as of its amplitude.

Previously published rotational frequencies of 2-, 4-, and 5-methyl oxazole and of 3- and 5-methyl isoxazole are reanalysed to obtain correct rotational constants (A constants in the first place) on the basis of hypothetically unsplit rotational centre frequencies. With these, r 0 -structural parameters of the respective methyl group could be determined, thus allowing us to consider the influence of methyl substitution on the 14 N-quadrupole coupling tensor and the structural relevance of the internal rotor angle given by torsional analysis. 14 N nuclear quadrupole hfs and IAM methyl torsional analyses were redone with the correct rotational constants.

Measurements of the 1 H spin-lattice relaxation time T 1 , the linewidth parameter T* 2 the second moment of 1 H NMR absorption, differential thermal analysis, and differential scanning calorimetry were performed on methylammonium tetrabromozincate(II) crystals from 58 to above 500 K. A solid-solid phase transition was located at 456 K. In the room temperature phase, 120° reorientational jumps of CH 3 and NH 3 + groups in the cation about its C -N bond axis were detected. In the high-temperature phase, the cations undergo overall reorientation as well as translational self-diffusion. The activation energy for the cationic self-diffusion was evaluated to be 18 kJ mol -1 .

The turbulence spectra of continuous fluids at large k are discussed. A basic difference in behaviour appears between conservative and dissipative systems. Only in the latter case can one eliminate higher order ultraviolet divergences.

The inviscid Burgers equation is studied with the Painleve analysis. A Bäcklund transformation is constructed. Then we give the symmetry generators. A two-dimensional case is also investigated.