The time-dependent variational method, which in many-body theory leads to the Hartree-Fock approximation, is here tested in quantum-mechanical models inspired by the physics of the inflationary universe. Some remarks about field-theoretic applications are also made.

We consider classical Hamiltonian systems in which there exist collective modes where the motion associated with each collective mode is describable by a collective coordinate. The formalism we develop is applicable to both continuous and discrete systems where the aim is to investigate the dynamics of kink or solitonlike solutions to nonlinear Klein-Gordon equations which arise in field theory and condensed-matter theory. We present a new calculational procedure for obtaining the equations of motion for the collective coordinates and coupled fields based on Dirac's treatment of constrained Hamiltonian systems. The virtue of this new (projection-operator) procedure is the ease with which the equations of motion for the collective variables and coupled fields are derived relative to the amount of work needed to calculate them from the Dirac brackets directly. Introducing collective coordinates as dynamical variables into a system enlarges the phase space accessible to the possible trajectories describing the system's evolution. This introduces extra solutions to the new equations of motion which do not satisfy the original equations of motion. It is therefore necessary to introduce constraints in order to conserve the number of degrees of freedom of the original system. We show that the constraints have the effect of projecting out the motion in the enlarged phase space onto the appropriate submanifold corresponding to the available phase space of the original system. We show that the Dirac bracket accomplishes this projection, and we give an explicit formula for this projection operator. We use the Dirac brackets to construct a family of canonical transformations to the system of new coordinates (which contains the collective variables) and to construct a Hamiltonian in this new system of variables. We show the equations of motion that are derived through the lengthy Dirac bracket prescription are obtainable through the simple projection-operator procedure. We provide examples that illustrate the ease of this projectionoperator method for the single-and multiple-collective-variable cases. We also discuss advantages of particular forms of the Ansatz used for introducing the collective variables into the original system.

%e derive a complete Hamiltonian formalism for a kink on a one-dimensional discrete lattice in which the position of the center of the kink appears as one of the canonical variables. Our method is a generalization to the discrete lattice of the method used in field theory to introduce the soliton as a canonical degree of freedom. The derivation is valid for a particle chain in a periodic potential when there exists a solitary-wave solution in the continuum limit. %e show that the discrete lattice is responsible for an adiabatic dressing of the kink and for spontaneous emission of phonons. In the limit where the effective length of the kink is much larger than the interparticle spacing the kink experiences the well-known periodic Peierls-Nabarro potential. In the case of a short kink, the discrete lattice causes the continuum kink configuration to be adiabatically dressed, leading to a renormalization of the Peierls-Nabarro potential and in turn to an enhancement of corresponding smallamplitude oscillatory frequency. In addition, we formally derive an equation that describes the radiation of phonons by the moving kink, an effect of the lattice discreteness.

We demonstrate that the reconstruction of the Au(l 11) surface can be interpreted in terms of a new type of misfit dislocations, namely, double-sine-Gordon-type dislocations. First, we motivate the applicability of this class of solitons to the reconstruction problem. Second, we describe the procedure we have used to construct the model unit cell containing the double-sine-Gordon quasi one-dimensional dislocations. Finally, comparison with experimental He-scattering results is established by computation of the corresponding diffraction pattern using a hard corrugated wall and the eikonal approximation.

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