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In the present work, we consider a variety of two-component, one-dimensionalstates in nonlinear Schrodinger equations in the presence of a parabolic trap,inspired by the atomic physics context of Bose-Einstein condensates. The use ofLyapunov-Schmidt reduction methods allows us to identify persistence criteriafor the different families of solutions which we classify as (m,n), inaccordance with the number of nodes in each component. Upon developing theexistence theory, we turn to a stability analysis of the differentconfigurations, using the Krein signature and the Hamiltonian-Krein index astopological tools identifying the number of potentially unstableeigendirections for each branch. A systematic expansion of suitably reducedeigenvalue problems when perturbing off of the linear limit permits us toobtain explicit expressions for the eigenvalues of each of the statesconsidered. Finally, when the states are found to be unstable, typically byvirtue of Hamiltonian Hopf bifurcations, their dynamics is studied in order toidentify the nature of the respective instability. The dynamics is generallyfound to lead to a vibrational evolution over long time scales.

Existence, Stability and Dynamics of Harmonically Trapped One-Dimensional Multi-Component Solitary Waves: The Near-Linear Limit