Fri, Apr 19
Applied Math Seminar
Anna Vainchtein, University of Pittsburgh
Supersonic fronts and pulses in a lattice with hardening-softening interactions.
3:00PM, Math 122
This talk is based on recent joint work with Lev Truskinovsky (ESPCI ParisTech). We consider a version of the classical Hamiltonian Fermi-Pasta-Ulam problem with nonlinear force-strain relation in which a hardening response is taken over by a softening regime above a critical strain value. We show that in addition to pulses (solitary waves) this discrete system also supports non-topological and dissipation-free fronts (kinks). Moreover, we demonstrate that these two types of supersonic traveling wave solutions belong to the same family. Within this family, solitary waves exist for continuous ranges of velocity that extend up to a limiting speed corresponding to kinks. As the kink velocity limit is approached from above or below, the solitary waves become progressively more broad and acquire the structure of a kink-antikink bundle. We investigate stability of the obtained solutions via Floquet analysis and direct numerical simulations. To motivate and support our study of the discrete problem we also analyze a quasicontinuum approximation with temporal dispersion. We show that this model captures the main effects observed in the discrete problem.
Mon, Apr 22
Algebra Seminar
Thomas Creutzig, Edmonton/Erlangen
Representation theory of affine VOAs
4:00PM, Zoom (please email achirvas@buffalo.edu)
An affine Lie algebra is a central extension of the loop algebra of a finite dimensional Lie algebra. The vacuum modules at any complex level \(k\) of the affine Lie algebra themselves carry an interesting algebraic structure, namely that of a vertex operator algebra (VOA). There presentation theory of affine VOAs is rather rich, e.g. suitable categories of modules form ribbon categories and there are exciting connections to geometry, quantum groups, physics and much more.
I will give an overview on the state of the art in this area.
Tue, Apr 23
Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #1
4:00PM
Title: 2023-24 Myhill Lecture #1
Wed, Apr 24
Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #2
4:00PM
Title: 2023-24 Myhill Lecture #2
Thu, Apr 25
Colloquium
Tomasz Mrowka, MIT
2023-24 Myhill Lecture #3
4:00PM
Title: 2023-24 Myhill Lecture #3
Thu, May 2
Colloquium
Dr Willy Hereman, Colorado School of Mines
Symbolic computation of solitary wavesolutions and solitons through homogenization of degree
4:00PM, Mathematics Building room 250
A simplified version of Hirota's method for thecomputation of solitary waves and solitons of nonlinear PDEs will be presented.The approach requires a change of dependent variable so that the transformedPDE is homogenous of degree in the new variable.
The resulting homogenous PDE does not have tobe quadratic and the method still applies if its bilinear form is not known.Solitons are then computed using a perturbation scheme involving linear andnonlinear operators. For soliton equations the scheme terminates after a finitenumber of steps. To illustrate the approach, solitons are computed for a classof fifth-order KdV equations due to Lax, Sawada-Kotera, and Kaup-Kupershmidt.
Homogenization of degree also allows one tofind solitary wave solutions of nonlinear PDEs that are not completelyintegrable. Examples include the Fisher and FitzHugh-Nagumo equations, and acombined KdV-Burgers equation. When applied to a wave equation with a cubicsource term, the method leads to a `bi-soliton' solution which describes thecoalescence of two wavefronts.
The method is largely algorithmic andimplemented in Mathematica. A demonstration of the software packagePDESolitonsSolutions will be given.
Fri, May 3
Applied Math Seminar
Willy Hereman, Colorado School of Mines
Symbolic computation of conservation laws of nonlinear partial differential equations.
3:00PM, Math 122
A method will be presented for the symbolic computation of conservation laws of nonlinear partial differential equations (PDEs) involving multiple space variables and time.
Using the scaling symmetries of the PDE, the conserved densities are constructed as linear combinations of scaling homogeneous terms with undetermined coefficients. The variational derivative is used to compute the undetermined coefficients. The homotopy operator is used to invert the divergence operator, leading to the analytic expression of the flux vector.
The method is algorithmic and has been implemented in Mathematica. The software is being used to compute conservation laws of nonlinear PDEs occurring in the applied sciences and engineering.
The software package will be demonstrated for PDEs that model shallow water waves, ion-acoustic waves in plasmas, sound waves in nonlinear media, and transonic gas flow. Equations featured in this talk include the Korteweg-de Vries and Zakharov-Kuznetsov equations..