Résumé:
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.
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