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Maxwell-Bloch and Nonlinear Schrodinger Systems with Nonzero Background
Başlık:
Maxwell-Bloch and Nonlinear Schrodinger Systems with Nonzero Background
Yazar:
Li, Sitai, author. (orcid)0000-0002-6078-8069
ISBN:
9780438049239
Yazar Ek Girişi:
Fiziksel Tanımlama:
1 electronic resource (213 pages)
Genel Not:
Source: Dissertation Abstracts International, Volume: 79-10(E), Section: B.
Advisors: Gino Biondini Committee members: Brian Hassard; John Ringland.
Özet:
This thesis is concerned with the study of two main types of nonlinear evolution equations of physical significance: (i) Maxwell-Bloch (MB) systems and (ii) nonlinear Schrodinger (NLS) type.
Maxwell-Bloch systems characterize the nonlinear resonant interactions between coherent light and an active optical medium. These systems exhibit interesting optical phenomena, such as self-induced transparency, superfluorescence, spontaneous radiation processes, and slow light. The nonlinear Schrodinger (NLS) equation arises as a physical model in water waves, optics, plasmas, Bose-Einstein condensates, and many other fields. In fact, it was shown that the NLS equation is a universal model for the evolution of the envelope of a weakly nonlinear dispersive wave train. In some regimes, MB systems and NLS-type equations are completely integrable, with an infinitely dimensional Hamiltonian structure, and with the existence of a Lax pair. As a result, various analytical methods can be applied to study their solutions. In particular, a nonlinear analog of the Fourier transform---the inverse scattering transform (IST)---is a powerful technique that can solve the initial value problem (IVP) and which can be used to derive exact solutions.
Chapter 1 discusses and derives the scalar integrable MB systems. The Lax pair and some properties of the MB system are also given. The IVP for this system is also presented, Then the IST for ZBC on the real line is formulated. In particular, a standard treatment is performed on the direct problem, the propagation of all scattering data is calculated, and the inverse problem is formulated in terms of a Riemann-Hilbert problem (RHP). The IST is done independently of the detuning function, which describes the distribution of atoms in the medium with respect to their frequency. Therefore, the resulting solutions and analysis cover a relatively broad range of physical scenarios.
Chapter 2 generalizes the IST formalism of Chapter 1 and formulates the IST for the scalar MB system with NZBC. In particular, the direct problem is now formulated on a genus-zero, two-sheeted Riemann surface. Similarly to the case of ZBC, the propagation of the scattering data and boundary data is computed explicitly, and for an arbitrary detuning function, The inverse problem is again formulated as a RHP. Thereafter, the soliton solutions and various limiting cases are solved analytically. Explicit expressions for general soliton solutions, rational solutions and periodic solutions are obtained, and are characterized in detail. Finally, the stability of solutions is studied and is connected to a physical setting.
Chapter 3 generalizes the IST to study the IVP for coupled MB systems with NZBC. The broad outline of the IST with NZBC is alike to the scalar case. However, there are some major departures, as well as additional complications, one of which is the problem of finding a complete set of analytic eigenfunctions in order to be able to carry out the inverse problem. Moreover, the structure of solutions of the coupled system is more complex than the scalar case.
Chapter 4 studies pure soliton solutions of the focusing NLS equation with ZBC. These solutions have been known for a long time, but a general description of their behavior was still lacking regarding degenerate soliton groups. The term degenerate soliton group is used here to indicate that some soliton velocities coincide. The classical description of soliton solutions only considers nondegenerate soliton solutions, i.e., the case in which all soliton velocities are distinct.
Chapter 5 extends the results of the previous chapter to the case of NZBC, and it provides a study of the interactions and long-time asymptotics of solitons and degenerate soliton groups. Unlike the case of ZBC, the focusing NLS equation with NZBC in general admits four distinct types of soliton solutions, depending on the relation between the soliton and the nonzero background. As a result, the long-time asymptotics of each type of soliton solutions is studied separately. The interactions between solitons are also more complicated than the case of ZBC, depending on whether the two interacting solitons are of the same type or be of to different types.
Chapter 6 studies the contribution to the solution originated from the radiation in the focusing NLS equation with NZBC, and also studies the similarities and differences between solutions produced by several other focusing NLS-type equations. All these focusing systems with NZBC are shown to exhibit the phenomenon called modulational instability (MI), known as Benjamin-Feir instability in water waves. Modulational instability is the phenomenon whereby a constant background is unstable to long-wavelength perturbations. The long-time asymptotics for the baseline model was recently studied before using the IST and the Deift-Zhou's nonlinear steepest descent method, and the leading order asymptotic behavior was computed explicitly. (Abstract shortened by ProQuest.).
Notlar:
School code: 0656
Konu Başlığı:
Tüzel Kişi Ek Girişi:
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Yer Numarası | Demirbaş Numarası | Shelf Location | Lokasyon / Statüsü / İade Tarihi |
---|---|---|---|
XX(681937.1) | 681937-1001 | Proquest E-Tez Koleksiyonu | Arıyor... |
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