Applied Mathematics and Mathematical Physics Seminar: feride tiglay
Topic
Integrable evolution equations on spaces of tensor densities and their peakon solutions
Speakers
Details
Abstract
In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study a variety of equations of interest in mathematical physics. I will describe how to extend his formalism using tensor densities and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bihamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and some results on blow-up as well as global existence of solutions. Time permitting, I will describe the peakon solutions for these equations.
In a pioneering paper V. Arnold presented a general framework within which it is possible to employ geometric and Lie theoretic techniques to study a variety of equations of interest in mathematical physics. I will describe how to extend his formalism using tensor densities and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. These two PDE possess all the hallmarks of integrability: the existence of a Lax pair formulation, a bihamiltonian structure, the presence of an infinite family of conserved quantities and the ability to write down explicitly some of its solutions. I will also talk about local well-posedness of the corresponding Cauchy problem and some results on blow-up as well as global existence of solutions. Time permitting, I will describe the peakon solutions for these equations.