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Riemann-Hilbert Problems For Nonlocal Reverse-Time Nonlinear Second-Order And Fourth-Order Akns Systems Of Multiple Components And Exact Soliton Solutions, Alle Adjiri

USF Tampa Graduate Theses and Dissertations

We first investigate the solvability of an integrable nonlinear nonlocal reverse-time six-component fourth-order AKNS system generated from a reduced coupled AKNS hierarchy under a reverse-time reduction. Riemann-Hilbert problems will be formulated by using the associated matrix spectral problems, and exact soliton solutions will be derived from the reflectionless case corresponding to an identity jump matrix. Secondly, we present the inverse scattering transform for solving a class of eight-component AKNS integrable equations obtained by a specific reduction associated with a block matrix spectral problem. The inverse scattering transform based on Riemann-Hilbert problems is presented along with a jump matrix taken to …

Orthogonal Polynomials With Respect To The Measure Supported Over The Whole Complex Plane, Meng Yang

USF Tampa Graduate Theses and Dissertations

In chapter 1, we present some background knowledge about random matrices, Coulomb gas, orthogonal polynomials, asymptotics of planar orthogonal polynomials and the Riemann-Hilbert problem. In chapter 2, we consider the monic orthogonal polynomials, $\{P_{n,N}(z)\}_{n=0,1,\cdots},$ that satisfy the orthogonality condition,

\begin{equation}\nonumber \int_\mathbb{C}P_{n,N}(z)\overline{P_{m,N}(z)}e^{-N Q(z)}dA(z)=h_{n,N}\delta_{nm} \quad(n,m=0,1,2,\cdots), \end{equation}

where $h_{n,N}$ is a (positive) norming constant and the external potential is given by

$$Q(z)=|z|^2+ \frac{2c}{N}\log \frac{1}{|z-a|},\quad c>-1,\quad a>0.$$

The orthogonal polynomial is related to the interacting Coulomb particles with charge $+1$ for each, in the presence of an extra particle with charge $+c$ at $a.$ For $N$ large and a fixed ``c'' this …