## Research Seminars in the School of Mathematical Sciences

The School of Mathematical Sciences host regular research seminars delivered by internal and external researchers on topics across mathematical sciences. Seminars take place on the Kevin Street campus and are open to all.

## Seminar 19/02/13: Classifying the long-run behaviour of globally stable differential equations subject to random perturbations

Classifying the long-run behaviour of globally stable differential equations subject to random perturbations

John Appleby

School of Mathematical Sciences, DCU

Tuesday 19 February 2013

1pm, Room KA3-011, DIT Kevin Street

Abstract:

In many applications, it transpires that the evolution of a system can be modelled satisfactorily by an autonomous ordinary differential equation, and that this system has a unique and globally stable steady state. However,it may be more realistic to presume that the system is also subjected to external random forces which are independent of the state. The question now is: what happens to the solutions of the randomly perturbed system? In the talk, we show that it is possible to classify entirely the asymptotic behaviour of scalar equations, as well as a significant subclass of multidimensional systems. To a large extent, it can be shown that there is an equation-independent threshold of noise intensity above which solutions change from being asymptotically stable to being unbounded. Extensions of these results are possible to a wide class of differential systems, including equations with delay, periodic equations, and examples of ordinary differential equations which possess globally stable limit cycles. It is also possible to capture precisely these asymptotic features by means of a single suitable implicit stochastic numerical method.

## Seminar 05/02/13: Dressing method in soliton theory

Dressing method in soliton theory

Tihomir Valchev

Dublin Institute of Technology

Tuesday 5 February 2012

1pm, Room KA3-011, DIT Kevin Street

Abstract:

A brief and elementary introduction to one of the most widely used methods for solving nonlinear integrable equations - the dressing method. Although the origins of the dressing method date back to some classical results of G. Darboux, it was Zakharov and Shabat who developed the dressing technique systematically at the beginning of the 70's. Since that time their approach became an essential working tool for finding special types of solutions (solitons in particular) to integrable equations. A main advantage of the dressing method is that it can be applied to multicomponent equations as naturally as to scalar ones while its drawback is that it is very sensitive to the boundary conditions imposed on solutions. We shall give certain examples of classical nonlinear equations that can be solved easily by using the dressing method.

## Seminar 13/11/12: Cyclic universe with an inflationary phase from a cosmological model with real gas quintessence

Cyclic universe with an inflationary phase from a cosmological model with real gas quintessence

Emil Prodanov

Dublin Institute of Technology

Tuesday 13 November 2012

1pm, Room KA3-022, DIT Kevin Street

## Seminar 29/11/12: Eigenfunction expansions associated with the one-dimensional Schroedinger equation

Eigenfunction expansions associated with the one-dimensional Schroedinger equation

Daphne Gilbert

Dublin Institute of Technology

Thursday 29 November 2012

1pm, Room KA3-011, DIT Kevin Street

## Seminar 11/10/12: Indefinite spectral problems and the HELP inequality

Indefinite spectral problems and the HELP inequality

Alex Kostenko

Vienna University

Thursday 11 October 2012

1pm, Room KA3-011, DIT Kevin Street

Abstract:

We study two problems. The first one is the similarity problem for indefinite Sturm--Liouville spectral problems. The second object is the so-called HELP inequality, a version of the classical Hardy-Littlewood inequality proposed by W.N. Everitt in 1971. Both problems are well understood when the corresponding Sturm-Liouville differential expression is regular. Our main main objective is to give criteria for both the validity of the HELP inequality and the similarity to a self-adjoint operator in the singular case. Namely, we establish new criteria formulated in terms of the behavior of the corresponding Weyl-Titchmarsh m-functions at zero and at infinity. As a biproduct of this result we show that both problems are closely connected. Next we characterize the behavior of m-functions in terms of coefficients and then these results enable us to reformulate the obtained criteria in terms of coefficients. Finally, we apply these results for the study of the two-way diffusion equation, also known as the time-independent Fokker-Plank equation.