Existence and uniqueness of the global conservative weak solutions to the rotation-Camassa–Holm equation

We study herein the Camassa–Holm-type equation, which can be considered as a model in the shallow water for the long-crested waves propagating near the equator with effect of the Coriolis force due to the Earth's rotation. This quasi-linear equation is nonlocal with higher-order nonlinearities compared to the classical Camassa–Holm equation. We establish the global existence and uniqueness of the energy conservative weak solutions in the energy space $H^1$ to this model equation.     

傅科摆的几何分析1)

傅科摆以最美的物理实验著称,但对傅科摆的解释一直是大学物理课程的难点。与牛顿力学采用科里奥利力论证傅科摆的传统方法相比,本文采用了几何方法来研究傅科摆,并指出傅科摆的转动是在球面上平移速度矢量的结果,最后给出了几何方法和牛顿力学解法的等价性证明。     

High-resolution infrared spectroscopy of CH281BrF near 8 μm: rovibrational analysis of the ν3 and ν8 fundamentals and resonances with the dark states 2ν5 and ν6 + ν9

The infrared spectrum of isotopically enriched CH₂⁸¹BrF was investigated in the ν₃ and ν₈ region between 1150 and 1370 cm^?1 at a resolution of 0.003 cm^?1. The ν₃ vibration of symmetry species A^′ gives rise to an a-/b-hybrid band with a-type predominance, while the ν₈ mode of A^″ symmetry produces c-type absorption. Due to the proximity of the band origins to those of closely lying overtones and combination bands, the v₃ = 1 and v₈ = 1 levels were found perturbed through Coriolis resonance by the v₅ = 2 (A^′) and v₆ = v₉ = 1 (A^″) states, respectively. The spectral analysis resulted in the identification of 3132 transitions (J^″ ≤ 98 and K_a^″ ≤ 14) for the ν₃ and 2958 transitions (J^″ ≤ 68 and K_a^″ ≤ 19) for the ν₈ bands. The assigned data were fitted using the Watson's A-reduction Hamiltonian in the I^r representation and the perturbation operators. Although no transitions belonging to the perturbers were observed, the band origins and excited state parameters for fundamentals and ‘dark states’ together with coupling terms for the ν₃/2ν₅ and ν₈/ν₆ + ν₉ dyads were determined.     

Microwave spectrum, tunneling motions, and quadrupole coupling hyperfine structure of ethylene diamine

New transitions are measured in the microwave spectrum of two conformers of the non-rigid ethylene diamine (CH₂NH₂-CH₂NH₂) molecule using both molecular beam Fourier transform and stark modulation spectroscopy. Ab initio calculations are carried out to obtain geometries and energies for several stationary points of the potential energy surface and to gain quantitative information on the tunneling paths along which the molecule tunnels. The microwave data are reproduced using an IAM-like approach which accounts for the rotational dependence of the tunneling splitting corresponding to the interconversion large amplitude motion. For one of the conformers, the results of this analysis are consistent with the ones obtained through ab initio calculations. Hyperfine patterns are analyzed with the help of a theoretical approach accounting for the large amplitude motion effects and for hyperfine matrix elements within and between tunneling sublevels.     

Dynamic motion of whirling rods with Coriolis effect

Longitudinal vibrations coupled with transverse vibrations of whirling rods are investigated. It is known that longitudinal and transverse vibrations are governed by second and fourth order differential equations, respectively. Due to the Coriolis effect, a system of equations that governs the longitudinal and transverse displacements will be constructed by coupling these two equations together. Solutions of the equations assume small oscillations of vibration being superimposed on the steady state of the whirling rod. Exact and approximate solutions are obtained from the proposed governing equations, where the approximate solutions on displacements and natural frequencies are acquired by neglecting the Coriolis effect. A proposed numerical scheme known as complete function collocation method is implemented to solve the governing equations coupled with longitudinal and transverse displacements. The approximate results on both longitudinal and transverse natural frequencies show that natural frequencies are decreasing while the angular velocity of the rod is increasing. Exact and numerical results on both longitudinal and transverse natural frequencies show that there are no predictable trends whether natural frequencies are increasing or decreasing while the angular velocity of the rod is increasing.     

关于转动弦的波动方程

数学物理方法的一些学习指导书或教科书上给出了两种转动弦的波动方程,但没有考虑Coriolis力的影响,其讨论方法存在不正确(竖直转动重弦)或不全面(水平转动轻弦)之处.重新讨论了两种转动弦的波动方程及其求解问题.     

有完整Coriolis力和热源影响下超长波的解析解

利用修正的Burger模式,采用行波解和泰勒级数展开法得到有完整Coriolis力和热源影响下超长波的解析解.得到描述非线性超长波的KdV和KdV-mKdV方程,并得到它的椭圆余弦波解、孤立波解和三角函数周期解.     

Gas phase electronic spectrum of propadienylidene C3H2

The rotationally resolved vibronic bands in the forbidden electronic transition of the cumulene carbene C₃H₂ have been observed in the gas phase by cavity ring down absorption spectroscopy through a supersonic planar plasma with allene as precursor. The band detected in the 16 223 cm⁻¹ region is a result of vibronic interaction and is assigned to a combination of a₁ and b₂ vibrations with a frequency around 2250 cm⁻¹. Another vibronic band near 15 810 cm⁻¹ has an unusual rotational structure because the K_a = 0-1 subband is absent. It is assigned to a combination of a₁ and b₁ vibrations, ∼1850 cm⁻¹, which borrow intensity from the near lying state due to a-type Coriolis coupling. A rotational analysis using a conventional Hamiltonian for an asymmetric top molecule yields molecular constants for the vibrational excited levels of the ùA₂ state, which were used for the determination of the geometry. The stronger transition of C₃H₂, measured in a neon matrix in the 16 161-24 802 cm⁻¹ range, was not detected. The reason for this is a short lifetime of the state, leading to line broadening.     

New measurements of the microwave spectrum of formamide

New measurements of the microwave spectrum of formamide have been obtained in the frequency range from 49 to 340 GHz using the microwave spectrometer at the Institute of Radio Astronomy of NASU, Kharkov, Ukraine. An analysis of the rotational spectra of the ground, v₁₂, v₉, v₁₁ and 2v₁₂ excited vibrational states of the main isotopic species as well as of the ground states of the ¹³C, ¹⁵N and ¹⁸O substituted species has been carried out using SPFIT/SPCAT programs. The analysis of a strong Coriolis interaction coupling between v₉, v₁₁ and 2v₁₂ vibrational states of formamide has been also fulfilled as well as the analysis of the quadrupole hyperfine structure of the observed transitions. For the first time the quadrupole coupling parameters for the excited vibrational states and for the ¹⁸O substituted species of formamide were determined.     

Some comments on the Coriolis attenuation problem

To explore the Coriolis attenuation problem we have carried out a schematici _13/2 rotor plus single quasi-particle band-mixing calculation. The results reveal that the calculations are largely insensitive towards the location of the Fermi energy near the low-K single particle states only, and therefore are incapable of taking into account the transition from ‘full’ decoupling to ‘partial’ decoupling as the Fermi level is increased. We trace the possible reasons for this insensitivity and find that this may be primarily due to thebcs approximation for calculating the quasiparticle energies.