Vector nonlinear Schrodinger systems with nonzero boundary conditions
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Scalar and vector nonlinear Schrödinger (NLS) equations are universal models for the evolution of weakly nonlinear dispersive wave trains. As such, they appear in many physical contexts, such as deep water waves, nonlinear optics, acoustics, Bose-Einstein condensation, etc. Many of these equations are also completely integrable infinite-dimensional Hamiltonian systems, and as such, they possess a remarkably rich mathematical structure. As a consequence, they have been the object of considerable research over the last fifty years. In particular, it is well known that for the integrable cases, the initial value problem can in principle be solved by the inverse scattering transform (IST), a nonlinear analogue of the Fourier transform. The actual implementation of the IST, however, is heavily dependent on the specific class of initial conditions (IC) considered. Quite a bit is known for problems in which the IC are sufficiently localized and tend to zero at infinity. On the other hand, problems in which the IC tend to constant, nonzero values are not as well-characterized. These kinds of problems are also of great applicative interest in all fields in which the NLS equation and its variants arise (e.g., optics, water waves, etc.). This dissertation is concerned with the vector nonlinear Schrödinger (VNLS) equation in both the defocusing and focusing cases with both zero and nonzero boundary conditions (ZBC and NZBC, respectively) at infinity. Specifically, this dissertation contains the following original contributions: (i) In Chapter 2, the symmetries of the Jost eigenfunctions and the norming constants are rigorously derived for the Manakov system (i.e., the 2-component VNLS equation) with ZBC, which was previously explored in 2003 by Ablowitz, Prinari, and Trubatch. In addition, a novel approach to reconstructing the entire scattering matrix in terms of scattering data is presented. In particular, the analytic scattering coefficients that had not been explicitly reconstruced in the 2003 work are found to be solutions of appropriate Riemann-Hilbert problems (RHP). (ii) In Chapter 3, the IST for the defocusing Manakov system with NZBC (originally presented in 2006 by Prinari, Ablowitz, and Biondini) is made more rigorous, and several additional results are derived. Importantly, precise conditions on the potential that guarantee analyticity of the Jost eigenfunctions and scattering coefficients are given; the behavior at the branch points is discussed; the discrete spectrum is classified completely using newly-found symmetries of the Jost eigenfunctions; the entire scattering matrix is reconstructed in terms of scattering data; the inverse problem is formulated in terms of an RHP; this RHP is shown to admit a unique solution; a closed-form expression for general soliton solutions is computed for the case when the analytic scattering coefficients have any number of simple zeros; and explicit solutions corresponding to double zeros of the analytic scattering coefficients are constructed explicitly. (iii) In Chapter 4, the general framework used for the defocusing Manakov system with NZBC (presented in Chapter 3) is generalized to develop the IST for the focusing Manakov system with NZBC. This is a completely new result. Moreover, the theory is substantially different from that of the defocusing case. As in Chapter 3, all results are proved rigorously, and the discrete spectrum is studied and classified. Compact, closed-form expressions are given for general soliton solutions corresponding to the case when the analytic scattering coefficients have any number of simple zeros. All the various types of one-soliton solutions are presented explicitly, and their behavior is discussed. (iv) In Chapter 5, the IST for the defocusing 3-component VNLS equation with NZBC is developed. A complete set of analytic eigenfunctions is rigorously derived for each fundamental domain of analyticity, using triangular decompositions of the scattering matrix. Many new results dealing with the analyticity of the minors of the scattering matrix are obtained. As in the previous chapters, the symmetries of the eigenfunctions and scattering coefficients are found, the discrete spectrum is discussed, and the inverse problem is formulated in terms of an RHP. Using this RHP, explicit soliton solutions are computed in certain, specific cases. Then, the approach to the IST used in this chapter is compared with the approach used for the defocusing Manakov system in Chapter 3 to show that the two approaches are equivalent. The results of Chapter 3 have been submitted for publication in SIAM J. Math. Anal., and those of Chapter 4 to Nonlinearity. Both are still under review at the time of this writing. The results of Chapter 5, while not yet definitive, represent a key step forward towards the complete solution of the problem and will likewise be published as soon as the work is complete.