Indiana University-Purdue University Indianapolis (IUPUI) / We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

Riemann-Hilbert-Probleme sind Randwertaufgaben für im Einheitskreis $\mathbb D$ holomorphe Funktionen $w$, deren Randwerte $w(t)$ auf gewissen Kurven $M_t$ liegen sollen. Ein Teil der Untersuchungen ist dem Fall explizit gegebener Kurven gewidmet. Dabei werden bekannte Resultate über glatte Kurven auf stetige Restriktionskurven erweitert, und die Existenz von Lösungen in gewissen Hardy-Räumen gezeigt. Die Eindeutigkeitsfrage führt auf ein Gegenbeispiel, das zugleich eine Vermutung aus einer Dissertation von Belch widerlegt. Der andere Teil der Untersuchungen ist dem klassischen Fall geschlossener Restriktionskurven gewidmet. Hier steht statt der Abschwächung von Glattheitsvoraussetzungen die Formulierung geeigneter Nebenbedingungen im Mittelpunkt. Die Abhängigkeit der Lösung von Zusatzbedingungen erweist sich als Verallgemeinerung des Verhaltens von Blaschkeprodukten. Für drei Interpolationpunkte kann charakterisiert werden, wann durch sie eine Lösung mit Windungszahl 1 verläuft, durch $k$ Interpolationspunkte wird die Existenz einer Lösung mit Windungszahl $k-1$ gezeigt.

We begin this dissertation by presenting a brief introduction to the theory of solitons and integrability (plus some classical methods applied in this field) in Chapter 1, mainly using the Korteweg-de Vries equation as a typical model. At the end of this Chapter a mathematical framework of notations and terminologies is established for the whole dissertation. In Chapter 2, we first introduce two specific matrix spectral problems (with 3 potentials) associated with matrix Lie algebras $\mbox{sl}(2;\mathbb{R})$ and $\mbox{so}(3;\mathbb{R})$, respectively; and then we engender two soliton hierarchies. The computation and analysis of their Hamiltonian structures based on the trace identity affirms that the obtained hierarchies are Liouville integrable. This chapter shows the entire process of how a soliton hierarchy is engendered by starting from a proper matrix spectral problem. In Chapter 3, at first we elucidate the Gauge equivalence among three types $u$-linear Hamiltonian operators, and construct then the corresponding B\"acklund transformations among them explicitly. Next we derive the if-and-only-if conditions under which the linear coupling of the discussed u-linear operators and matrix differential operators with constant coefficients is still Hamiltonian. Very amazingly, the derived conditions show that the resulting Hamiltonian operators is truncated only up to the 3rd differential order. Finally, a few relevant examples of integrable hierarchies are illustrated. In Chapter, 4 we first present a generalized modified Korteweg-de Vries hierarchy. Then for one of the equations in this hierarchy, we build the associated Riemann-Hilbert problems with some equivalent spectral problems. Next, computation of soliton solutions is performed by reducing the Riemann-Hilbert problems to those with identity jump matrix, i.e., those correspond to reflectionless inverse scattering problems. Finally a special reduction of the original matrix spectral problem will be briefly discussed.

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 can be a small perturbation of the Gaussian weight. The polynomial $P_{n,N}(z)$ can be characterized by a matrix Riemann--Hilbert problem \cite{Ba 2015}. We then apply the standard nonlinear steepest descent method \cite{Deift 1999, DKMVZ 1999} to derive the strong asymptotics of $P_{n,N}(z)$ when $n$ and $N$ go to $\infty.$ From the asymptotic behavior of $P_{n,N}(z),$ we find that, as we vary $c,$ the limiting distribution behaves discontinuously at $c=0.$ We observe that the mother body (a kind of potential theoretic skeleton) also behaves discontinuously at $c=0.$ The smooth interpolation of the discontinuity is obtained by further scaling of $c=e^{-\eta N}$ in terms of the parameter $\eta\in[0,\infty).$ To obtain the results for arbitrary values of $c$, we used the ``partial Schlesinger transform'' method developed in \cite{BL 2008} to derive an arbitrary order correction in the Riemann--Hilbert analysis. In chapter 3, we consider the case of multiple logarithmic singularities. The planar orthogonal polynomials $\{p_n(z)\}_{n=0,1,\cdots}$ with respect to the external potential that is given by $$Q(z)=|z|^2+ 2\sum_{j=1}^lc_j\log \frac{1}{|z-a_j|},$$ where $\{a_1, a_2, \cdots, a_l\}$ is a set of nonzero complex numbers and $\{c_1, c_2, \cdots, c_l\}$ is a set of positive real numbers. We show that the planar orthogonal polynomials $p_{n}(z)$ with $l$ logarithmic singularities in the potential are the multiple orthogonal polynomials $p_{{\bf{n}}}(z)$ (Hermite-Pad\'e polynomials) of Type II with $l$ measures of degree $|{\bf{n}}|=n=\kappa l+r,$ ${\bf{n}}=(n_1,\cdots,n_l)$ satisfying the orthogonality condition, $$ \frac{1}{2\ii}\int_{\Gamma}p_{{\bf{n}}}(z) z^k\chi_{{\bf{n}}-{\bf{e}}_j}(z)\dd z=0, \quad 0\leq k\leq n_j-1,\quad 1\leq j\leq l,$$ where $\Gamma$ is a certain simple closed curve with counterclockwise orientation and $$ \chi_{{\bf{n}}-{\bf{e}}_j}(z):= \prod_{i=1}^l(z-a_i)^{c_i }\int_{0}^{\overline{z}\times\infty}\frac{\prod_{i=1}^l(s-\bar{a}_i)^{n_i+c_i}}{(s-\bar{a}_j)\ee^{zs}}\,\dd s. $$ Such equivalence allows us to formulate the $(l+1)\times(l+1)$ Riemann--Hilbert problem for $p_n(z)$. We also find the ratio between the determinant of the moment matrix corresponding to the multiple orthogonal polynomials and the determinant of the moment matrix from the original planar measure.

Stuttgart, Univ., Diss., 2008.

We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

Riemann-Hilbert-Probleme sind Randwertaufgaben für im Einheitskreis $\mathbb D$ holomorphe Funktionen $w$, deren Randwerte $w(t)$ auf gewissen Kurven $M_t$ liegen sollen. Ein Teil der Untersuchungen ist dem Fall explizit gegebener Kurven gewidmet. Dabei werden bekannte Resultate über glatte Kurven auf stetige Restriktionskurven erweitert, und die Existenz von Lösungen in gewissen Hardy-Räumen gezeigt. Die Eindeutigkeitsfrage führt auf ein Gegenbeispiel, das zugleich eine Vermutung aus einer Dissertation von Belch widerlegt. Der andere Teil der Untersuchungen ist dem klassischen Fall geschlossener Restriktionskurven gewidmet. Hier steht statt der Abschwächung von Glattheitsvoraussetzungen die Formulierung geeigneter Nebenbedingungen im Mittelpunkt. Die Abhängigkeit der Lösung von Zusatzbedingungen erweist sich als Verallgemeinerung des Verhaltens von Blaschkeprodukten. Für drei Interpolationpunkte kann charakterisiert werden, wann durch sie eine Lösung mit Windungszahl 1 verläuft, durch $k$ Interpolationspunkte wird die Existenz einer Lösung mit Windungszahl $k-1$ gezeigt.

Optimal control problems can in many cases become complicated and difficult to solve. One particular class of difficult control problems to solve are singular control problems. Standard methods for solving optimal control are discussed showing why those methods are difficult to apply to singular control problems. Then standard methods for solving singular control problems are discussed including why the standard methods can be difficult and often impossible to apply without having to resort to numerical techniques. Finally, an alternative method to solving a class of singular optimal control problems is given for a specific class of problems.

Este trabalho trata de problemas de contorno definidos no plano. O problema central desta tese é chamado Problema de Riemann-Hilbert, o qual pode ser descrito como segue. Seja L um campo vetorial complexo não singular definido em uma vizinhança do fecho de um aberto simplesmente conexo do plano com fronteira suave. O Problema de Riemann-Hilbert para o campo L consiste em obter uma solução para a equação Lu = F(x, y, u) no aberto em estudo, sendo dada uma função F mensurável. Pede-se também que a solução tenha extensão contínua até a fronteira e que satisfaça lá uma condição adicional; trabalha-se aqui no contexto das funções Hölder contínuas. Foram obtidos resultados para o problema acima no caso em que L pertence a uma classe de campos hipocomplexos. O caso clássico conhecido é quando o campo vetorial é o operador de Cauchy-Riemann, ou, mais geralmente, quando é um campo elítico. / This work deals with boundary problems in the plane. The central problem in this thesis is the so-called Riemann-Hilbert problem, which may be described as follows. Let L be a non-singular complex vector field defined on a neighborhood of the closure of a simply connected open subset of the plane having smooth boundary. The Riemann-Hilbert problem for the vector field L consists in finding a solution to the equation Lu = F(x, y, u) on the open set under study, where the given function F is measurable. It is also required that the solution have a continuous extension up to the boundary and satisfy an additional condition there. Results were obtained for the above problem when L belongs to a class of hypocomplex vector fields. The well-known classical case is the one in which the vector field under study is the Cauchy-Riemann operator, or more generally when it is an elliptic vector field.

Στην παρούσα εργασία ασχολούμαστε με μεθόδους κατασκευής λύσεων για μη γραμμικές μερικές διαφορικές εξώσεις (ΜΔΕ) εξέλιξης, δηλαδή εξισώσεις που περιγράφουν μια φυσική κατάσταση που εξελίσσεται χρονικά, και διακρίνονται σε γραμμικές και μη γραμμικές. Για την επίλυση των γραμμικών ΜΔΕ εξέλιξης υπάρχει η μέθοδος του μετασχηματισμού Fourier. Για τις μη γραμμικές ΜΔΕ εξέλιξης δεν υπάρχει κάποια γενική μέθοδος κατασκευής λύσεων. Πολλές απ’ αυτές, έχουν την ιδιότητα να επιδέχονται ειδικές λύσεις που ονομάζονται σολιτόνια. Βασικό χαρακτηριστικό των σολιτονίων είναι η «ελαστική» αλληλεπίδρασή τους. Πρώτοι οι Zabusky και Kruskal ανακάλυψαν το 1965 ότι η εξίσωση των Korteweg και De Vries (KdV) επιδέχεται σολιτονική λύση. Σχεδόν αμέσως οι Gardner, Greene, Kruskal και Miura [1967,1974] βρήκαν μια μέθοδο κατασκευής σολιτονικής λύσης για την εξίσωση KdV. Η μέθοδος βασίζεται στην λογική της σκέδασης και της αντίστροφης σκέδασης. Η μέθοδος της αντίστροφης σκέδασης, λειτουργεί ανάλογα με αυτή του μετασχηματισμού Fourier για τις γραμμικές, και αποτελεί το κύριο μέρος αυτής της εργασίας. Ειδικότερα: Στο πρώτο κεφάλαιο, παρουσιάζουμε παραδείγματα γραμμικών εξισώσεων εξέλιξης σε μία χωρική διάσταση, καθώς και λύσεις αυτών. Στη συνέχεια, αναζητούμε σολιτονικές λύσεις για τις μη γραμμικές ΜΔΕ εξέλιξης και κλείνουμε με ένα παράδειγμα μη γραμμικής ΜΔΕ εξέλιξης στις δύο χωρικές διαστάσεις. Στο δεύτερο κεφάλαιο, δείχνουμε πώς μπορούμε να κατασκευάσουμε λύσεις προβλημάτων αρχικών τιμών (ΠΑΤ) για γραμμικές εξισώσεις εξέλιξης, με χρήση του μετασχηματισμού Fourier. Στη συνέχεια, γίνεται εφαρμογή της μεθόδου της αντίστροφης σκέδασης στην κατασκευή λύσεων για μη γραμμικές ΜΔΕ εξέλιξης. Στο τρίτο κεφάλαιο, γίνεται εφαρμογή της μεθόδου της αντίστροφης σκέδασης στο ΠΑΤ για την εξίσωση KdV. Για κατάλληλη επιλογή της αρχικής συνθήκης διαπιστώνουμε ότι η KdV επιδέχεται σολιτονικές λύσεις. Συγκεκριμένα, επιλέγουμε αρχικές συνθήκες που εξελίσσονται χρονικά σε σολιτονική, 2-σολιτονική και 3-σολιτονική λύση. Τέλος, παρουσιάζουμε ένα πρόγραμμα σε περιβάλλον Mathematica που κατασκευάζει πολυσολιτονική λύση για την εξίσωση KdV. Το τέταρτο κεφάλαιο αφιερώνεται στα ζεύγη Lax, τα οποία είναι ζεύγη γραμμικών εξισώσεων εξέλιξης. Αυτό που τα χαρακτηρίζει είναι ότι, η συνθήκη συμβατότητας αυτών είναι η εξίσωση εξέλιξης που μας ενδιαφέρει. Σε αυτό βασίζεται και η μέθοδος των Ablowitz, Kaup, Newell και Segur (AKNS), για την κατασκευή λύσεων μη γραμμικών εξισώσεων εξέλιξης. Εφαρμόζουμε την μέθοδο AKNS στην εξίσωση KdV για να κατασκευάσουμε σολιτονικές λύσεις. Στο πέμπτο και τελευταίο κεφάλαιο, ασχολούμαστε με την αναδιατύπωση ενός ΠΑΤ ως πρόβλημα Riemann-Hilbert. Επιπλέον, δείχνουμε πώς συνδέεται ένα πρόβλημα αντίστροφης σκέδασης με ένα πρόβλημα Riemann-Hilbert, θεωρώντας την εξίσωση KdV. Τέλος, αναφερόμαστε στην σύνδεση προβλημάτων αρχικών-συνοριακών τιμών με το πρόβλημα Riemann-Hilbert και κάνουμε μια επισκόπιση στη σύγχρονη βιβλιογραφία και παρουσιάζουμε πρόσφατα αποτελέσματα σε αυτή την κατεύθυνση. / In this master thesis our subject is to construct solutions for nolinear partial differential evolution equations (PDEs), which are equations that describe a physical model that evolves in time, and can be either linear or nonlinear. For solving linear PDEs we use the Fourier Transform (FT), while for nonlinear PDEs a general method for constructing solutions does not exist. Many of them admit special kind of solutions that are called solitons. A basic property of solitons, is that they interact in an elastic way. In 1965, Zabusky and Kruskal were the first to discover that the Korteweg & de Vries (KdV) equation admits a soliton solution. Straightforward Gardner, Greene, Kruskal and Miura [1967, 1974] found a method to contruct a soliton solution for the KdV equation. This method is based on the Inverse Scattering Transform (IST). The IST is the nonlinear FT- analogue, and a big part of our work is devoted to this method. Particularly: In the first chapter, we introduce some examples of linear evolution equations in one spatial dimension, and their solutions. We then construct soliton solutions for nonlinear evolution PDEs and an example in 2 spatial dimensions is considered. The second chapter deals with Initial Value Problems (IVP) and their solution construction via the FT. We also apply the IST to construct solutions for nonlinear evolution PDEs. In the third chapter, we consider KdV as an example of an evolution equation that is integrable under the IST, by the knowledge of the initial distribution of the solution. For a specific choise of the initial condition we establish that KdV equation admits soliton solutions. Especially, we choose initial conditions that evolve in time to 1-soliton, 2-soliton and multi-soliton solution. Finally, we present a program with Mathematica that constructs multi-soliton solution for the KdV. The lax pair for a nonlinear evolution equation is introduced in the fourth chapter. Lax pairs are pairs of linear PDEs and, often, their compatibility condition is the nonlinear equation we study. The method produced by Ablowitz, Kaup, Newell and Segur (AKNS), for constructing solutions for nonlinear evolution equations, is based on Lax pairs. We apply this method to KdV. The last chapter refers to Riemann Hilbert (RH) problems and their connection with the Inverse Scattering problem. We use KdV to show this connection. Finally, we mention how an Initial and Boundary Value Problem (IBVP) and an RH problem are connected. A quick review of recent results is considered.