Binary mixtures with a time-dependent variational approach: new families of breathers in two-component condensates

Abstract

Our work is organized as follows. In chapter 1, we give a short historic and background about the nonlinear Schr odinger equation (NLSE) and its solitonic solutions. In chapter 2, we present the Balian-V en eroni time dependent variational principle, which is the main tool to derive the Gross-Pitaevskii equation (GPE) and its generalizations in a mean eld framework. Since these equations are highly nonlinear, they require special analytic tools. We present the general formalism of the Darboux transformation method and the Lax pair method. For readers who are not familiar with these methods, we present a simple example. In chapter 3, we focus on two component condensates where we nd solitonic solutions of the coupled Gross-Pitaevskii equations (CGPE). By transforming our model to a Manakov system via similarity transformation and employing Darboux transformation with zero seed, we observe that the introduction of an external trap leads to sudden shoots up in the atomic density indicating onset of dynamical instability. We pursue our analysis in chapter 4 by nding another type of solitons, namely the Peregrine solitons. The Darboux transformation is used in two cases. The symmetric case with the same seed solutions and the nonsymmetric case. One also observes the onset of dynamical instability as the frequency of the harmonic trap is varied. By a speci c choice of the spectral parameters, we show that these solitons may be stabilized. In chapter 5, we generalize our approach by letting free the spectral parameters. We nd general families of solitonic solutions parametrized by the spectral parameters. We derive not only the Peregrine solitons found previously, but also the standard families of Ma and Akhmediev breathers as well as new general breathers and rogue waves. In all these cases, we show that, by modulating the trap frequency, we are be able to stabilize the solitons against dynamical instability. In the last part of our work, we gather some conclusions and perspectives. 1