Math 348: Topics in Analysis
AbstractThe focus of the course is to teach modern steepest descent techniques that allow the asymptotic solution of integrable differential and difference equations that are completely integrable in terms of explicit formulae. The term ''asymptotic'' refers to either the behavior of an evolution system after long time (i.e. after the initial transient) or in a semiclassical limit (in which the system involves some small coefficient such as the dispersion coefficient in the case of the Korteweg de-Vries equation or the ''Planck constant'' in the nonlinear Schroedinger equation). The problem of solving the system in question is cast as a Riemann-Hilbert problem (prior knowledge not required) and the steepest descent method allows us to replace it with a much simpler one that may be solved explicitly. The method is rigorous, error estimates are obteined. Time permititng, some of the significant advances that these methods have effected on the theory of orhtogonal polynomials and random matrices will be presented.PrerequisitesThe course will contain an ample introduction to integrable systems and Riemann-Hilbert problems, so that no previous knowledge in these subjects will be required. No prior knowledge of terms used in this description: Korteweg de-Vries equation, Planck constant, nonlinear Schroedinger equation, orhtogonal polynomials, random matrices will be necessary. The methods described are based on analyticity. Cauchy's theorem, the Cauchy integral formula, poles and residues, multivalued functions with branches and branch-cuts (mainly the square root and the logarithm), will be relied on. A course in complex variables that covers these topics is a prerequisite.Mail comments and suggestions concerning this site to dgs-math@math.duke.edu Last modified: |
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dgs-math@math.duke.edu
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 Mathematics
DepartmentDuke University, Box 90320 Durham, NC 27708-0320 | |
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