The systematic analysis begins by considering the dark-bright (DB)-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component BECs. The interaction between two DB solitons in a homogeneous condensate and at the presence of the trap are both considered. Our analytical approximation relies in a Hamiltonian perturbation theory, which leads to an equation of motion of the centers of DB-soliton interacting pairs. Employing this equation, we demonstrate the existence of robust DB-soliton molecules, in the form of stationary two- and three-DB-soliton states. Also the equilibrium distance of the constituent solitons and the corresponding oscillation frequencies are found semianalytically, where the latter corresponds to the characteristic anomalous modes' eigenfrequencies that we numerically computed via a so called Bogoliubov-de Gennes (BdG) linearization analysis. Those studies are discussed in Chapter 2.

Then, we extend our studies to the dynamics of dark-bright (DB) solitons in binary BECs at finite temperature using a system of two-coupled dissipative GPs. We show that the effect of the bright soliton is to partially stabilize dark solitons against temperature-induced dissipation, thus providing longer lifetimes in Chapter 3.

Furthermore, the dark-dark (DD) solitons as a prototypical coherent structure that emerges in two-component BECs are studied and are connected to dark-bright (DB) solitons via SO(2) rotation. We obtained their beating frequency and their frequency of oscillation inside a parabolic trap. They are identified as exact periodic orbits in the Manakov limit of equal inter- and intra- species nonlinearity strengths with and without the trap and we showcase the persistence of such states upon weak deviations from this limit. Also we investigated in detail the effect of the deviation from the Manakov case by considering different from unity scattering length ratios in Chapter 4.

Next, we revisited Hamiltonian eigenvalue problems that typically arise in the linearization around a stationary state of a Hamiltonian nonlinear PDE. Also we presented a overview of the known facts for the eigenvalue counts of the corresponding unstable spectra. In particular, we focused on a straightforward plan to implement finite-dimensional techniques for locating this spectrum via the singular points of the meromorphic Krein Matrix and illustrated the value of the approach by considering realistic problems for recently observed experimentally multivortex and multisoliton solutions in atomic Bose-Einstein condensates in Chapter 5.

In the two dimensional scenario, we also examine the stability and dynamics of vortices under the effect of dissipation used as a simplified model for the inclusion of the effect of finite temperatures in atomic BECs, which enables an analytical prediction that can be compared directly to numerical results in Chapter 6.

In all the above studies, our analytical prediction from the equation of motion are in good agreement with the numerical results from the BdG analysis.

]]>Because the *p*-parts and global series are closely related, the result above follows from a series of local results concerning the *p*-parts. In particular, we give an explicit recurrence relation on the coefficients of the *p*-parts, which allows us to extend the results of Chinta, Friedberg, and Gunnells [9] to all _ and *n*. Additionally, we show that the *p*-parts of Chinta and Gunnells [10] agree with those constructed using the crystal graph technique of Brubaker, Bump, and Friedberg [4, 5] (in the cases when both constructions apply).