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The number state method is used to study soliton bands for three anharmonic quantum lattices: i) The discrete nonlinear Schrödinger equation, ii) The Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these systems is assumed to have f -fold translational symmetry in one spatial dimension, where f is the number of freedoms (lattice points). At the second quantum level (n = 2) we calculate exact eigenfunctions and energies of pure quantum states, from which we determine binding energy (E b ), effective mass (m * ) and maximum group velocity (Vm) of the soliton bands as functions of the anharmonicity in the limit f → ∞. For arbitrary values of n we have asymptotic expressions for E b , m * , and Vm as functions of the anharmonicity in the limits of large and small anharmonicity. Using these expressions we discuss and describe wave packets of pure eigenstates that correspond to classical solitons.

The discrete self-trapping (DST) equation describes a coupled system of anharmonic oscillators that can be quantised in a remarkably simple manner. Here the DST system is used to describe the relationship between quantum and classical descriptions of local modes of vibration in a molecule.

We develop the theory of generalized Weierstrass σ-and ℘-functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the ℘-functions, a proof that the coefficients of the power series expansion of the σ-function are polynomials of coefficients of the defining equation of the curve, and the derivation of two addition formulae.

We review work on the Discrete Nonlinear Schrödinger (DNLS) equation over the last two decades.

We consider a hierarchy of the natural type Hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of 2 × 2 matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Using the method of variable separation we provide the integration of the systems in classical mechanics conctructing the separation equations and, hence, the explicit form of action variables. The quantisation problem is discussed with the help of the separation variables.

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We show for the first time that highly localized in-plane breathers can propagate in specific directions with minimal lateral spreading in a model 2-D hexagonal non-linear lattice. The lattice is subject to an on-site potential in addition to longitudinal nonlinear inter-particle interactions. This study investigates the prediction that stable breather-like solitons could be formed as a result of energetic scattering events in a given layered crystal and would propagate in atomic-chain directions in certain atomic planes. This prediction arose from a long-term study of previously unexplained dark lines in natural crystals of muscovite mica.63.20.Pw, 63.20.Ry, 03.40.Kf.The response of a nonlinear 2D atomic lattice when embedded in a surrounding 3D lattice is presently of much interest in connection with transport phenomena in layered crystals. Two examples of such crystals are the copper-oxide based high temperature superconductors and the potassium based silicate muscovite mica. Fortunately the optical transparency of the latter allows the study of energetic events at the atomic level. This is possible in mica because of tracks made visible by impurities of transient defects, through triggered solid-state phase transitions. A study of these tracks arising from nuclear scattering events led to the suggestion that energy could be transported over large distances through the crystal by some sort of energetic intrinsic localized mode (ILM) on the lattice [1]. The interesting and novel aspect of this prediction was that these ILMs would travel along the crystal axes and remain localized in both longitudinal and transverse directions with little or no lateral spreading. This is despite the fact that the 2D lattice has full hexagonal symmetry. This one-dimensional behavior in a two-dimensional lattice was called quasi-onedimensional (QOD) and the resulting ILMs were called "quodons" [2].The purpose of this present letter is to demonstrate numerical simulations of a mica-like model which support the QOD conjecture. Our intention is not to provide a detailed model of the mica system at this stage, but merely to present a simplified model which maintains much of the overall qualitative features of the mica system and yet can be easily studied. The model is general enough to suggest the possibility of QOD effects in other hexagonal crystal structures, and even other geometries such as square lattices.Muscovite mica has a layered structure. A conspicuous feature of this structure is the mono-atomic planes of *