No abstract

No abstract

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.

No abstract

We use a recently developed projection-operator approach to derive the equation of motion for the center of mass of a discrete sine-Gordon (SG) kink, where the center of mass of the kink is represented by the collective variable X. We calculate the small-oscillation frequency of the discrete SG kink trapped inside the Peierls-Nabarro (PN) potential well using an ansatz that introduces the collective variable X into the system and which incorporates discreteness into the kink's "shape mode. " We obtain essentially exact agreement between molecular-dynamics simulation and theory.We show that when the small-oscillation PN frequency is calculated using the bare ground state of the SG lattice, such as in Ishimori and Munakata's use of the Keener and McLaughlin perturbation theory, or in our own bare treatment, that the square of the PN frequency is approximately a factor of 2 smaller than values obtained from simulation, even when the kink size is such that discreteness eft'ects are small. In particular, we show that the ratio of the curvature at the bottom of the dressed PN well to the curvature at the bottom of the bare PN well is approximately a factor of 2 for lo & 3.We also numerically calculate a first-order dressing for the bare kink, and use this first-order dressed ground state to calculate the small-oscillation PN frequency which agrees with simulation to within 5%. We briefly discuss large-amplitude oscillations of the kink in the PN well.

No abstract

No abstract

We present a complete Hamiltonian treatment of a kink with an internal degree of freedom, namely the double sine-Gordon (DSG) kink. In this formalism we assign two canonical coordinates and their associated momenta to describe the motion of the center of mass of the DSG kink and the relative motion of its two subkinks. We show that the canonical coordinate representing the separation of the two subkinks describes a nonlinear oscillatory degree of freedom. Consequently, the DSG kink behaves like a "molecule" (4~kink) comprised of two "atoms" (each of a single 2~kink) held together by a nonlinear potential. As an application of our formalism, we obtain the solutions for the nonlinear internal motion of the DSG in the absence of the radiation field.