Experimental studies of the superconductive properties of fullerides are briefly reviewed. Theoretical calculations of the electron-phonon coupling, in particular for the intramolecular phonons, are discussed extensively. The calculations are compared with coupling constants deduced from a number of different experimental techniques. It is discussed why ${\mathrm{A}}_{3}$${\mathrm{C}}_{60}$ are not Mott-Hubbard insulators, in spite of the large Coulomb interaction. Estimates of the Coulomb pseudopotential ${\mathrm{\ensuremath{\mu}}}^{\mathrm{*}}$, describing the effect of the Coulomb repulsion on the superconductivity, as well as possible electronic mechanisms for the superconductivity, are reviewed. The calculation of various properties within the Migdal-Eliashberg theory and attempts to go beyond this theory are described.

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Superconductivity in fullerides

A new method valid for highly dispersive and highly nonlinear water waves is presented. It combines a time-stepping of the exact surface boundary conditions with an approximate series expansion solution to the Laplace equation in the interior domain. The starting point is an exact solution to the Laplace equation given in terms of infinite series expansions from an arbitrary z-level. We replace the infinite series operators by finite series (Boussinesq-type) approximations involving up to fifth-derivative operators. The finite series are manipulated to incorporate Pade approximants providing the highest possible accuracy for a given number of terms. As a result, linear and nonlinear wave characteristics become very accurate up to wavenumbers as high as kh = 40, while the vertical variation of the velocity field becomes applicable for kh up to 12. These results represent a major improvement over existing Boussinesq-type formulations in the literature. A numerical model is developed in a single horizontal dimension and it is used to study phenomena such as solitary waves and their impact on vertical walls, modulational instability in deep water involving recurrence or frequency downshift, and shoaling of regular waves up to breaking in shallow water.

A new Boussinesq method for fully nonlinear waves from shallow to deep water

The interaction of weakly nonlinear long internal gravity waves is studied. Weak interactions occur when the wave phase speeds are unequal; this case includes that of a head-on collision. It is shown that each wave satisfies a Korteweg-de Vries equation, and the main effect of the interaction is described by a phase shift. Strong interactions occur when the wave phase speeds are nearly equal although the waves belong to different modes. This case is described by a pair of coupled Korteweg-de Vries equations, for which some preliminary numerical results are presented.

Weak and strong interactions between internal solitary waves

The most abundant fullerene molecule, C60, has just the right combination of size, chemical stability, and activity to serve as the building block for a large variety of solids with fascinating properties. After a short overview of the related carbon compounds, first the structures of the C60 molecule and the pure crystalline C60 are summarized. Experimental and theoretical works on the vibrational and the electronic properties of the molecule and the solid are reviewed. Next, some of the relevant concepts widely used in solid state phyics, like electron-phonon and electron-electron interactions, Hund's rule and Jahn-Teller distortion are discussed in the context of doped fullerides. The structural and electronic instabilities of the conducting C60 compounds are reviewed in detail. Finally, a few open questions are discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/64/5/202/pdf

Electronic properties of doped fullerenes

In order to model the evolution of a solitary wave near an obstacle or over an uneven bottom, the long-wave equations including curvature effects are introduced to describe the deformation and fission of a barotropic solitary wave passing over a shelf or an obstacle. The numerical results obtained from these equations are shown to be in good agreement with an analytical model derived by Germain (1984) in the framework of a generalized shallow-water theory, and with experimental results collected in a large channel equipped with a wave generator. Given the initial conditions, i.e. amplitude of the incident solitary wave, water depth in the deep region, and height of the shelf or the barrier, it is possible to predict the amplitude and number of the transmitted solitary waves as well as the amplitude of the reflected wave, and to describe the shape of the free surface at any time.

Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle

We consider a head-on collision between two solitary waves on the surface of an inviscid homogeneous fluid. A perturbation method which in principle can generate an asymptotic series of all orders, is used to calculate the effects of the collision. We find that the waves emerging from (i.e. long after) the collision preserve their original identities to the third order of accuracy we have calculated. However a collision does leave imprints on the colliding waves with phase shifts and shedding of secondary waves. Each secondary wave group trails behind its primary, a solitary wave. The amplitude of the wave group diminishes in time because of dispersion. We have also calculated the maximum run-up amplitude of two colliding waves. The result checks with existing experiments.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112080000262

On head-on collisions between two solitary waves

A force-constant model previously developed for ${\mathrm{C}}_{60}$ is extended to ${\mathrm{C}}_{70}$. Using the force constants determined for the ${\mathrm{C}}_{60}$ molecule, the vibrational frequencies for the ${\mathrm{C}}_{70}$ molecule are calculated and are identified with the irreducible representations of the ${\mathit{D}}_{5\mathit{h}}$ symmetry group, according to which the vibrational modes transform. A comparison of the predictions of this model with the experimentally reported values for the Raman- and infrared-actaive modes shows good agreement overall. The effect of doping by alkali-metal atoms is taken into account by modifying the force constants in order to produce a good fit to the available data for doped ${\mathrm{C}}_{60}$. The model is then used to predict the frequencies of the Raman- and infrared-active modes in alkali-metal-doped ${\mathrm{C}}_{70}$.

Force-constant model for the vibrational modes in C60.

Collisions between two solitary waves are investigated using a numerical scheme. The phase shifts and maximum amplitude of a collision are checked with a corresponding perturbation calculation and compared with the available experiments. We found a wave train trailing behind each of the emerging solitary waves from a head-on collision. The properties of the wave train are in agreement with those of the perturbation solution. After the collision, the solitary waves recover almost all of their original amplitude for the length of time in our calculation. However, the difference (less than 2 % of their original value) persists and accounts for the energy residing in the wave train.

Collisions between two solitary waves. Part 2. A numerical study

A ninth‐order perturbation solution describing the solitary wave at the interface of a two‐fluid system is found by means of a symbolic manipulation software package. The coefficients in the general solution are evaluated for 37 values of the density and depth ratios σ and R, yielding the first nine terms in the series expansions for the wave profile, wave speed, mass, and potential energy in powers of the wave amplitude a. A maximum in the ninth‐order partial sum for the wave speed is found in each case for which the sum is judged to be an acceptable approximation to the infinite series for all a less than or equal to Af=(R−(σ)1/2 )/(1+(σ)1/2). Furthermore, this maximum occurs for a value of a very close to the value of Af. In all 37 cases, the [5,4] Pade approximant for the wave speed has a maximum, and in almost all cases the maximum occurs close to Af. No maximum is found in the ninth‐order partial sums for mass or potential energy in the range of values of a for which these sums are judged to be valid.

Rida M. Mirie

Papers

On head-on collisions between two solitary waves

Force-constant model for the vibrational modes in C60.

Collisions between two solitary waves. Part 2. A numerical study

Internal solitary waves in a two‐fluid system

Internal solitary waves and their head-on collision. Part 1