## 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 26/09/13: Nonlinear Stationary Phase

Nonlinear Stationary Phase

Spyros Kamvissis

Thursday 26 September 2013

1pm, Room KA3-011, DIT Kevin Street

Abstract:

We consider the stability of the stationary and the periodic Toda lattice under a "short range" perturbation. It is an old result that in the first case the perturbed lattice asymptotically reduces to a finite system of solitons. In the periodic case, however it approaches a modulated lattice (plus a finite system of solitons) that we describe explicitly. Our method relies on the equivalence of the inverse spectral problem to a matrix Riemann-Hilbert factorisation problem: in the first case it is defined in the complex sphere while in the periodic case it is defined in a hyperelliptic curve. We prove our result by applying (and generalising) the so-called nonlinear stationary phase method for Riemann-Hilbert problems.

## Seminar 18/04/13: Direct and inverse spectral problems for 2-dimensional Hamiltonian systems

Direct and inverse spectral problems for 2-dimensional Hamiltonian systems

Harald Woracek

Vienna University of Technology

Thursday 18 April 2013

1pm, Room KA3-011, DIT Kevin Street

## 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 11/04/13: Analyticity of rotational travelling water waves

Analyticity of rotational travelling water waves

Prof. Joachim Escher

Gottfried Wilhelm Leibniz Universität, Hannover, Germany

Thursday 11 April 2013

1pm, Room KA3-011, DIT Kevin Street

Abstract:

It is shown that the streamlines beneath a periodic water wave travelling over a flat bed with wavespeed which exceeds the horizontal velocity of all fluid particles are real-analytic curves if the vorticity function is merely integrable to some power. This regularity implies the following symmetry result: If it is a priori known that the streamlines attain their global minimum on a vertical line and that the wave is locally one-sided monotonic, then there is a minimal period such that within that period the wave possesses exactly one crest and one trough and it is symmteric with respect to the line perpendicular through the crest.

## 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.

### More Articles...

- Seminar 29/11/12: Eigenfunction expansions associated with the one-dimensional Schroedinger equation
- Seminar 13/11/12: Cyclic universe with an inflationary phase from a cosmological model with real gas quintessence
- Seminar 11/10/12: Indefinite spectral problems and the HELP inequality
- Seminar 04/10/12: Seminar Elliptic variational problems with non-local operators