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Showing 1 to 7 of 7 for “"riemann-hilbert problem"”.
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The Importance of the Riemann-Hilbert Problem to Solve a Class of Optimal Control Problems
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 …
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A Riemann-Hilbert approach to scattering on a one-dimensional semi-infinite crystal
… crystal is analyzed using the method of the Riemann-Hilbert Problem for analytic functions. We proceed in the framework of the limiting absorption principle. Our formulas allow to find the scattered field at any point of the crystal. In particular, we obtain an explicit formula for the …
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On the discrete nonlinear Schrödinger equation with PT-Symmetry
… discrete Lax operator. Based on the formulated Riemann-Hilbert problem, we derive the one- and two-soliton solutions to the nonlocal Ablowitz-Ladik equation. Finally, we prove the completeness relation for the associated Jost solutions. Based on this, we derive the expansion formula over the …
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Orthogonal Polynomials On An Arc Of The Unit Circle With Respect To A Generalized Jacobi Weight: A Riemann-Hilbert Method Approach
… polynomials as solutions of a 2X2 matrix Riemann-Hilbert problem, which extends to the unit circle the original Riemann-Hilbert characterization for orthogonal polynomials on the real line, first discovered by Fokas, Its, and Kitaev. In order to extricate the behavior of the polynomials …
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Combinatorics Of The Hermitian One-Matrix Model
… interpretation involving graphs embedded in Riemann surfaces. Generating functions for these graphs in the case of an even potential have been studied by many authors. The case of a cubic potential has also been studied. Using string equations, we construct "valence independent" formulas for …
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A Riemann-Hilbert Approach to the Kissing Polynomials
… known as the Kissing Polynomials through Riemann-Hilbert techniques. The Kissing Polynomials are a family of non-Hermitian orthogonal polynomials, which are orthogonal with respect to the complex weight function $\exp(i\omega z)$ over the interval $[-1,1]$, where $\omega>0$. Although they …
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A continuum limit of a finite discrete nonlinear Schroedinger system
… that is, to formulate a variational maximization problem whose solution characterizes the limiting behavior of the inverse spectral solutions of the discrete systems. We convert the maximization problem into a Riemann-Hilbert problem on the unit circle. The "one-gap" ansatz is studied for times …