In this dissertation, some of the major obstacles facing the design of a future multistageplasma-based linear collider (LC) are addressed. Plasma-based acceleration (PBA) is being considered for future LC designs because of large acceleration gradients that may dramatically reduce the size and cost. A critical challenge for PBA is ensuring the quality, i.e., the energy spread and emittance, of the witness beam is maintained. This requires transporting the witness beam in a stable manner even when there is an offset between the drive and witness beams, both in the plasma and between stages. When the tightly focused, high charge electron beams required for LCs are matched to the transverse wakefield, the space charge forces of the beam move the plasma ions significantly within the transit time of the beam. This ion motion, perturbs the wakefields which can cause degradation to the beam quality. In this dissertation, the acceleration of electron beams in nonlinear wakefields driven by electron beams is investigated using both theoretical models and particle-in-cell (PIC) simulations. Cases where there are misalignments between the drive and witness beam, both with and without ion motion are considered. Novel ideas to help address these issues are provided.

The existing theory for describing the hosing instability is extended to regimes relevantto future experiments and ultimately the LC regime. An azimuthal mode decomposition is employed to solve for the fields at the plasma sheath boundary, improving the hosing theory so that it provides better agreement with PIC simulations. Another issue, that has largely been unstudied, is beams with asymmetrical transverse sizes. The same azimuthal decomposition method can be used to characterize the wakefields created by such asymmetric beams.

Methods to mitigate the hosing instability are discussed and investigated. There havebeen many studies recently with different methods to detune the resonance between the beam and plasma channel and damp the instability. These are based on varying the betatron frequency (oscillation frequency of the centroid of the beam) or focusing force along the bunch. The idea of using a sufficient energy chirp on the witness beam that will eventually be corrected while accelerating because it underloads the plasma wake is discussed, as well as the idea of using an asymmetric drive beam that causes the focusing force to vary along the witness beam, both of which will cause the witness beam hosing to be damped.

The issue of plasma ion motion is addressed. The ion density distributions and resultingwakefields from tightly focused electron beams are described and a model for how the emittance evolves in these ion density profiles when the beam is asymmetric or has an offset is developed. It was found that when the drive beam induces ion motion before the witness beam, it can fully eliminate the hosing and realign the witness beam, but at the cost of large emittance growth in a uniform plasma. Using PIC simulations to study the LC regime, where ion motion is significant, it is possible to adiabatically match the beams in the presence of ion motion using plasma density ramps while essentially eliminating any transverse misalignment of the witness beam and maintaining the beam emittance, enabling stable transport of high quality electron beams for future PWFA-LCs. A single stage for which the witness electron beam is realigned, emittance growth is limited to a few percent, the energy spread is limited to less than a percent, and there is 50 percent drive-to-witness energy transfer efficiency is presented.