| This thesis is made up of two parts, each containing two articles, and consists of a study, from a mathematical point of view, of baryonic systems in 3+1 and 2+1 dimensions using effective Skyrme-type Lagrangians.;The first part is about soliton scattering in the Skyrme model in 3+1 dimensions. In article 1 we use Manton's method of truncation of degrees of freedom to obtain the force induced by the kinetic term of the Skyrme Lagrangian between two well separated Skyrmions, using the product ansatz. The product ansatz has been used to parameterize the pair of Skyrmions for large impact parameter, permitting therefore the use of free Skyrmion collective coordinates to describe each particle. This enables us to compute some classical trajectories of Skyrmion scattering. In article 2, we study the implications of the kinetic term found in article 1 on the exclusive scattering of physical nucleons, obtained by semi-classical quantization of the appropriate collective coordinates describing the Skyrmions. In order to do this, we use the approximation of "variation of constants" and we are able to treat the problem without numerical integration. This method treats the interaction as a perturbation on the free dynamics. We compute here the nucleon-nucleon scattering for slow initial speeds, and a large impact parameter for certain cases of spin and isospin polarization.;The second part of the thesis studies "bound" states in Skyrme-type Lagrangians in 2+1 dimensions. In article 3 we present string-like solutions for the baby Skyrme model in 2+1 dimensions, using topological considerations and continuous deformations of "Skyrme matter". This model is the analog of the Skyrme model consisting of a kinetic term, a Skyrme-type term and a mass term. In article 4 we study the properties of a model obtained from the baby-Skyrme model but removing the term quadratic in derivatives. This model is symmetric under volume preserving diffeomorphisms in 2+1 dimensions, and possesses compact solitons whose profiles can be computed analytically. We take advantage of that fact by computing the quantum rotational spectrum of the baby-Skyrmion while taking into account the soliton deformations under centrifugal forces. We also construct from the string type solution, a physical state of finite baryon number whose shape is similar to an annulus, and compute its energy. We find it degenerate with that of the baby-Skyrmion and the straight string. We conclude with some indications on the scattering of baby-Skyrmions. (Abstract shortened by UMI.). |
