A density matrix formalism is developed for anelastic (mechanical) relaxation in crystalline materials with point defects characterized by elastic dipoles. The time-dependent approach to equilibrium of the strain response under the action of a constant applied stress is deduced. The formalism parallels the one used in nuclear magnetic relaxation. The anelastic relaxation time is determined as a function of the parameters occurring in the defect hopping term in the Hamiltonian. This term is responsible for the dissipation of the anelastic ‘potential’ energy into the host lattice. In a lengthy concluding section, the following aspects are discussed point by point: the advantages of the formalism presented, its scope and special cases; the physical implications of the expression obtained for the relaxation time; the similarities and differences between magnetic relaxation and anelastic relaxation, etc.

We briefly review the nonlinear dynamics of diverse physical systems which can be described in terms of moving curves and surfaces. The interesting connections that exist between the underlying differential geometry of these systems and the corresponding nonlinear partial differential equations are highlighted by considering classic examples such as the motion of a vortex filament in a fluid and the dynamics of a spin chain. The association of the dynamics of a non-stretching curve with a hierarchy of completely integrable soliton-supporting equations is discussed. The application of the surface embeddability approach is shown to be useful in obtaining such connections as well as exact solutions of some nonlinear systems such as the Belavin-Polyakov equation and the inhomogeneous Heisenberg chain.

Using classical differential geometry, we discuss the phenomenon of anholonomy that gets associated with a static and a moving curve. We obtain the expressions for the respective geometric phases in the two cases and interpret them. We show that there is a close connection between anholonomy and nonlinearity in a wide class of nonlinear systems.

The Gross–Pitaevskii equation (GPE) describing the evolution of the Bose–Einstein condensate (BEC) order parameter for weakly interacting bosons supports dark solitons for repulsive interactions and bright solitons for attractive interactions. After a brief introduction to BEC and a general review of GPE solitons, we present our results on solitons that arise in the BEC of hard-core bosons, which is a system with strongly repulsive interactions. For a given background density, this system is found to support both a dark soliton and an antidark soliton (i.e., a bright soliton on a pedestal) for the density profile. When the background has more (less) holes than particles, the dark (antidark) soliton solution dies down as its velocity approaches the sound velocity of the system, while the antidark (dark) soliton persists all the way up to the sound velocity. This persistence is in contrast to the behaviour of the GPE dark soliton, which dies down at the Bogoliubov sound velocity. The energy–momentum dispersion relation for the solitons is shown to be similar to the exact quantum low-lying excitation spectrum found by Lieb for bosons with a delta-function interaction.

We present a unified formulation to investigate solitons for all background densities in the Bose–Einstein condensate of a system of hard-core bosons with nearest-neighbour attractive interactions, using an extended Bose–Hubbard lattice model. We derive in detail the characteristics of the solitons supported in the continuum version, for the various cases possible. In general, two species of solitons appear: A nonpersistent (NP) type that fully delocalizes at its maximum speed and a persistent (P) type that survives even at its maximum speed. When the background condensate density is nonzero, both species coexist, the soliton is associated with a constant intrinsic frequency, and its maximum speed is the speed of sound. In contrast, when the background condensate density is zero, the system has neither a fixed frequency, nor a speed of sound. Here, the maximum soliton speed depends on the frequency, which can be tuned to lead to a cross-over between the NP-type and the P-type at a certain critical frequency, determined by the energy parameters of the system. We provide a single functional form for the soliton profile, from which diverse characteristics for various background densities can be obtained. Using mapping to spin systems enables us to characterize, in a unified fashion, the corresponding class of magnetic solitons in Heisenberg spin chains with different types of anisotropy.