This dissertation concentrates on the existence, stability and dynamical properties of nonlinear waves in Bose-Einstein condensates (BECs) trapped in doublewell potentials (DWPs). The fundamental model of interest will be the nonlinear Schrödinger equation, the so-called Gross-Pitaevskii (GP) equation, contributed to the well-established mean-field description of BECs. In this context of the GP equation with DWP, a Galerkin-type few-mode approach provides us a powerful handle towards studying the stationary states and predicting the bifurcation diagram including the occurrence of spontaneous symmetry breaking (SSB). Such method and the corresponding phenomena are discussed based on a prototypical quasi-1D model in Chapter 2.

The systematic analysis progresses by considering various modified models, starting with the ones involving different interatomic interactions, e.g., collisionally inhomogeneous interactions, long-range interactions, and competing of short- and long-range interactions. We observe how the basic SSB bifurcation structure persists or is appropriately modified in the presence of these interactions in Chapter 3.

We also extend the study to multi-component systems, including nonlinearly coupled two-component settings and F = 1 spinor BECs (genuinely three-component settings) confined in DWPs, where besides the one-component stationary states, combined states involving two or three components appear as well, and novel SSB phenomena emerge within them. Finally the trapped stationary modes of a twodimensional (2D) GP equation with a symmetric four-well potential are explored, providing the picture of SSB in the fundamental 2D setting. These various systems are studied in Chapter 4 - 6.

In all models, our analytical predictions based on the few-mode approximation are in excellent agreement with the numerical results of the full GP equations.