Monday, July 22
11:15 AM

IP1
A Steepest-Descent/Stationary Phase Type Method for Oscillatory Riemann-Hilbert Problems. Applications to Integrable Models in Mathematical Physics

An astonishing variety of integrable problems in mathematics, applied mathematics and mathematical physics can be solved in terms of an associated Riemann-Hilbert (RH) problem. Classical examples,such as radiative equilibrium, often fall under the general rubric of Wiener-Hopf factorization theory. More modern examples include the Korteweg-de Vries equation, the nonlinear Schrodinger equation, integrable models in statistical mechanics, Painleve equations, and random matrix models.

Frequently the coefficients in the RH problem are oscillatory or exponentially growing/decreasing, and depend,moreover,on external parameters such as space x, time t, dispersion coefficient D, etc. The interesting problem is to analyze the behavior of the RH problem, and hence the asymptotic behavior of the solution of the underlying physical system, as the coefficients (x and t, say) become large or small (say, D).

The speaker will discuss a new nonlinear steepest descent/stationary phase type method to compute the behavior of such oscillatory/exponentially diverging RH problems. The method, which is rigorous and algorithmic, proceeds in analogy with the classical method of steepest descent/stationary phase, by locating the points of stationary phase,deforming contours, etc. The method has now been applied and extended by various authors to obtain the asymptotic behavior of a very wide spectrum of integrable models.

Percy Alec Deift
Courant Institute of Mathematical Sciences
New York University

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