热点大瓜

Fri, May 2

Applied Math Seminar
Alex Bivolcic (UB), Applied Math Seminar, MATH250
Dispersive Shock Waves and Integrability of Modulation Equations for the Kadomtsev-Petviashvili Equation


3:00PM, MATH250

Nonlinear wave phenomena comprise an important class of problems in mathematical physics. Remarkably, many of the governing equations, which arise as universal models in a variety of physical settings, are completely integrable infinite-dimensional Hamiltonian systems. Accordingly, such equations have a rich mathematical structure. This talk is concerned with the study of the Kadomtsev-Petviashvili (KP) equation, which comes in two variants, referred to respectively as the KPI and KPII equations.聽聽The KP equation is a universal model that describes the evolution of weakly dispersive, nonlinear wave trains in two spatial dimensions. It arises in many physical contexts, including shallow water waves, plasmas, acoustics, optics, and Bose-Eistain condensates. It is a completely integrable infinite-dimensional Hamiltonian system, and possesses an infinite number of conserved quantities. This talk is split into two parts:

Part I: Two-Dimensional Reductions of the Whitham Modulation System for the KP Equation. Various two-dimensional reductions of the KP-Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the KP equation, are studied and characterized. Three different reductions are considered, corresponding to modulations that are independent of x, independent of y, and of t (i.e., stationary), respectively. Each of these reductions still describe dynamic, two-dimensional spatial configurations,
since the modulated cnoidal wave generically has a nonzero speed and a nonzero slope in the xy plane.
In all three of these reductions, the properties of the resulting systems of equations are studied. It is shown that the resulting reduced system is not integrable unless one enforces the compatibility of the system with all conservation of waves equations (or considers a reduction to the harmonic or soliton limit). In all cases, compatibility with conservation of waves yields a reduction in the number of dependent variables to two, three and four, respectively. As a by-product of the stationary case, the Whitham modulation system for the Boussinesq equation is also explicitly obtained.

Par tII: Mach Reflection and Expansion of Two-Dimensional Dispersive Shock Waves Generated by Wedge-Type Initial Conditions. The oblique collisions and dynamical interference patterns of two-dimensional dispersive shock waves are studied numerically and analytically via the temporal dynamics induced by wedge-shaped initial conditions for the KPII equation. Various asymptotic wave patterns are identified, classified, and characterized in terms of the incidence angle and the amplitude of the initial step, which can give rise to either subcritical or supercritical configurations, including the generalization to dispersive shock waves of the Mach reflection and expansion of viscous shocks and line solitons. An eight-fold amplification of the amplitude of an obliquely incident flow on a wall at the critical angle is demonstrated. Applications of the results include bore interactions in geophysical fluid dynamics.

Mon, May 5

Algebra Seminar
Claudiu Raicu, University of Notre Dame
Polynomial functors and stable cohomology
4:00PM, 250 Math Building

The theory of polynomial representations of the general linear group goes back to the thesis of Issai Schur at the turn of the 20th century. Such representations include the tensor, symmetric, and exterior powers of a vector space, and have been completely classified in the work of Schur when the underlying field is the complex numbers. While there has been significant progress since the work of Schur, the story over a field of positive characteristic remains largely unknown. In my talk I will describe some novel stabilization results for sheaf cohomology, and explain their connection to the study of polynomial representations / functors. This is based on joint work with Keller VandeBogert.