On Multi-component Nonlinear Schrödinger Equation with PT-Symmetry
The aim of this thesis is to develop the inverse scattering method for multi-component generalisations of nonlocal reductions of the nonlinear Schrödinger equation with parity- time (PT)-symmetry related to symmetric spaces. This includes: the spectral properties of the associated Lax operator, Jost solution, the scattering matrix, the minimal sets of scattering data and the fundamental analytic solutions. We also derive the completeness relations of Jost solutions which are related to the spectral decomposition of the operator L(λ). As main examples, we use the Manakov vector Schrödinger equation (related to A.III-symmetric spaces) and the multi-component nonlinear Schrödinger equations of Kulish-Sklyanin type (related to BD.I-symmetric spaces). Furthermore, the one- and two-soliton solutions, subject to different reductions, are obtained by using an appropriate modification of the Zakharov-Shabat dressing method. It is shown that the multi-component nonlinear Schrödinger equations of these types allow both regular and singular soliton configurations. Also, we show that the inverse scattering method can be viewed as a generalised Fourier transform. Then, based on Wronskian relations for the fundamental analytic solutions, completeness relations for the so-called “squared solutions” are derived. Based on these completeness relations, expansions of generic functions from the class of admissible potentials over the complete set of “squared solutions” are derived. This allows one to describe the class of nonlinear evolutionary equations integrable by the inverse scattering method for a given spectral problem and the recursion operator, generating the corresponding integrable hierarchy. Then, we briefly outline the construction of the infinite set of integrals of motion. Finally, we study and determine the potentially essential condition for creating soliton solutions in vector nonlocal nonlinear Schrödinger equation (related to A.III-symmetric spaces). We study two cases of symmetric and asymmetric potentials. Numerical simulations are used to solve the system of equations and illustrate our analytical results.
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