# Applied Mathematics Seminar

## Seminars in 2016

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### Seminars in 2016, Semester 2

Wednesday November 30

Andrew Kels (University of Queensland)

Wednesday November 9

Peter Clarkson (University of Kent)

Wednesday October 19

Herbert Huppert (Cambridge University/University of NSW)

Volcanic eruptions, fluid intrusions and false similarity solutions

Wednesday September 21

Kenji Kajiwara (Kyushu University)

An approach to discrete differential geometry from integrable systems

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Classical differential geometry is one of the sources of integrable systems which dates back to the 19th century, as seen in the classical works of Backlund and Darboux on the construction of transformations of surfaces. Recently discretization of geometric objects such as surfaces and curves preserving underlying properties has attracted considerable attention from various points of view. In this talk, we explain an approach from integrable systems by choosing two examples:

(1) deformation of smooth/discrete curves: isoperimetric deformation in the Euclidean plane by (discrete) mKdV equation (2) discrete holomorphic function: a discrete power function and the Painlevé VI equation

In the first half, starting from some historical remarks and basic ideas of integrable systems and their discretization, we explain how the hierarchies of integrable systems naturally arise in the context of deformation theory of plane/space curves. Then we demonstrate a formulation of discrete deformation preserving its integrable nature. In the second half, we present an outline of ongoing project with Nalini Joshi, Nobutaka Nakazono and Yang Shi. We explain how the Painlevé equations arise in the theory of discrete holomorphic functions, and give an explicit formula of a discrete power function by Agafonov and Bobenko in terms of the hypergeometric tau function of the Painlevé VI equation.

Wednesday September 14

Shane Keating (University of New South Wales)

Flavours of Baroclinic Instability in the Global Ocean

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The formation of turbulent eddies in the ocean is primarily driven by baroclinic instability, which releases available potential energy stored in sloping density surfaces. The details of the local shear and stratification profile can give rise to qualitatively distinct flavours of baroclinic instability, however. In particular, the presence of a surface density gradient can have a strong impact on the necessary conditions for baroclinic instability and the resulting linear and nonlinear dynamics. In this talk, the effect of a surface density gradient on the formation, vertical structure, and transport properties of eddies and waves is studied in an "oceanic" version of the classical Charney problem. The analysis is carried out in the context of a simple model flow that permits a systematic study of the competing effects of surface buoyancy, planetary vorticity, and baroclinic shear in terms of two nondimensional parameters that emerge naturally from the problem: the Charney-Green number relating the strength of the planetary potential vorticity gradient to the surface buoyancy gradient, and the Phillips supercriticality quantifying the stability of the mean shear profile. Different flavours of baroclinic instability are represented by different regimes in parameter space. Within this framework, hydrographic temperature and salinity profiles are used to form a global atlas of baroclinic instability in the ocean, and regional and seasonal patterns are discussed.

Wednesday August 24

Mathieu Desroches (INRIA Sophia Antipolis)

Canards in piecewise-linear slow-fast systems

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In this talk I will review results from the 1990s as well as more recent work on how to "reconstruct" canard dynamics using piecewise-linear (PWL) slow-fast systems. I will first present the main elements necessary to obtain a canard explosion in planar PWL systems, depending on the number of linearity zones considered. Then I will focus on three-dimensional systems with two slow variables, showing how to recover folded-node dynamics in the PWL context and how to construct canard-induced mixed-mode oscillations. Finally, I will briefly mention one example of 3D PWL system with two fast variables displaying square-wave bursting with spike-adding canard explosion. Throughout the talk, some emphasis will be given to application to neuronal dynamics.

Wednesday July 27

Philippe Guyenne (University of Delaware)

The surface signature of internal waves in the ocean

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Based on a Hamiltonian formulation of a two-layer ocean, we consider the situation in which the internal waves are treated in the long-wave regime while the surface waves are described in the modulation regime. We derive an asymptotic model for surface-internal wave interactions, in which the nonlinear internal waves evolve according to a KdV equation while the smaller-amplitude surface waves propagate at a resonant group velocity and their envelope is described by a linear Schrodinger equation. In the case of an internal soliton of depression, for small depth and density ratios of the two layers, the Schrodinger equation is shown to be in the semi-classical regime and thus admits localized bound states. This leads to the phenomenon of trapped surface modes, which propagate as the signature of the internal wave, and thus it is proposed as a possible explanation for bands of surface roughness above internal waves in the ocean. A few numerical simulations taking oceanic parameters into account are also performed to illustrate this phenomenon. This is joint work with Walter Craig and Catherine Sulem.

Wednesday June 29

Yury Stepanyants (University of Southern Queensland)

Modulational instability of quasi-harmonic wave-trains in rotating fluids.

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The problem of modulational stability of quasi-monochromatic wave-trains propagating in a media with a double dispersion is revised. A typical example of double-dispersion medium is a rotating fluid which provides both the small-scale Boussinesq dispersion and large-scale Coriolis dispersion. We derive the nonlinear Schrodinger equation (NLSE) from the basic set of hydrodynamic equations in the long-wave limit taking into account the Coriolis force. For unidirectional waves propagating in one direction only the considered set of equations reduces to the Gardner-Ostrovsky equation which is applicable within a finite range of wavenumbers. It is shown that the narrow-band wave-trains are modulationally stable for relatively small wavenumbers k < kc and unstable for k > kc, where kc is some critical wavenumber. The derived NLSE is applicable to any wavenumbers up to zero. The detailed analysis of coefficients of the NLSE is presented for different signs of dispersive coefficients.

### Seminars in 2016, Semester 1

Wednesday May 25

Ngamta (Natalie) Thamwattana, (University of Wollongong)

Applications of calculus of variations for modelling proteins, nucleic acids, polymers and carbon nanostructures

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Calculus of variations is a classical technique, involving the optimization of a mathematical energy associated to the problem under consideration. It has applications in various fields, such as economics, physics, aeronautics, mechanical engineering and sport equipment design. In the setting of smooth functions finding a critical point of the energy is analogous to the first derivative test of undergraduate calculus, and the condition of a stationary point gives a corresponding differential equation called the Euler-Lagrange equation associated to the energy. In this talk, I will present some of my work on using calculus of variations to model various molecular structures, including proteins, nucleic acids, polymers and carbon nanostructures.

Wednesday May 18

Darren Engwirda, (Massachusetts Institute of Technology)

Towards optimal simplicial mesh generation in two- and three-dimensions

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Unstructured mesh generation is a key component in various computational modelling and simulation tasks, including scientific computing, computer graphics and animation, and various data visualisation applications. In this talk, I present a range of techniques for the construction of triangular and tetrahedral meshes for planar, surface and volumetric domains. I focus on two key areas: (i) the development of "provably-good" algorithms based on "restricted" Delaunay tessellations, and (ii) the use of local topological and geometrical optimisation to achieve quasi-optimal mesh adaptation and improvement. In addition to describing a class of practically useful algorithms, I also aim to expose several open theoretical problems worthy of additional investigation. A range of results are presented using our unstructured meshing package JIGSAW, including the generation of isotropic tessellations for various two- and three-dimensional geometries, and the use of solution-adaptive re-meshing strategies to construct high-quality anisotropic meshes for PDE-based models.

Wednesday May 11

Chris Lustri (University of Sydney)

Applications of Exponential Asymptotic Methods

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The utility of asymptotic series expansions has been long-established within applied mathematics for providing approximations to exact solutions in some asymptotic limit. However, these methods are typically unable to capture behaviour that is exponentially-small in the limit, irrespective of how many terms of the series one chooses to take. Behaviour on this scale is described as lying "beyond-all-orders".

This talk will be divided into two parts. In the first part, I will discuss how exponential asymptotic methods may be used to obtain information about behaviour that occurs on an exponentially-small scale, and in particular, how such methods uncover behaviour known as the Stokes Phenomenon. In the second part of the talk, I will discuss applications of these methods to problems arising in fluid dynamics, and the study of discrete systems.

Wednesday May 4

Bronwyn Hajek, (University of South Australia)

Viscous extensional flow and Symmetry analysis for reaction-diffusion equations

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In the first part of the talk, I will describe some problems in viscous extensional flow, including honey dripping from an upturned spoon and the stretching of a viscous thread as its ends are pulled apart. Of interest here, is the determination of the time place where the thread ÒpinchesÓ, and in particular, the relationship of these to the initial shape of the drop or thread. These types of problems have many applications such as the drawing of glass and polymer fibres for optical microscopy or microelectrodes.

In the second part of the talk, I will discuss the application of symmetry techniques to reaction-diffusion equations. These types of equations can be used to describe behaviour in many types of systems such as in population dynamics, cell biology, heat transfer and flow in porous media. Symmetry analysis is a useful technique which allows the construction of analytic solutions, enabling the importance of various system parameters to be determined easily.

Wednesday April 27

Peter van Heijster, (Queensland University of Technology)

A geometric approach to stationary defect solutions in one space dimension

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In this talk, we consider the impact of a small jump-type spatial heterogeneity on the existence of stationary localized patterns in a system of partial differential equations in one spatial dimension, i.e., defined on $\mathbb{R}$. This problem corresponds to analyzing a discontinuous and non-autonomous $n$-dimensional system, $\scriptsize\dot{u}=\left\{ \begin{array}{ll} f(u),& t\leq0,\\ f(u)+\varepsilon g(u),& t>0, \end{array}\right.$ under the assumption that the unperturbed system, i.e., the $\varepsilon \to 0$ limit system, possesses a heteroclinic orbit $\Gamma$ that connects two hyperbolic equilibrium points (plus several additional nondegeneracy conditions). The unperturbed orbit $\Gamma$ represents a localized structure in the PDE setting. We define the (pinned) defect solution $\Gamma_\varepsilon$ as a heteroclinic solution to the perturbed system such that $\lim_{\varepsilon \to 0} \Gamma_\varepsilon = \Gamma$ (as graphs). We distinguish between three types of defect solutions: trivial, local, and global defect solutions. The main goal of this manuscript is to develop a comprehensive and asymptotically explicit theory of the existence of local defect solutions. We find that both the dimension of the problem as well as the nature of the linearized system near the endpoints of the heteroclinic orbit $\Gamma$ have a remarkably rich impact on the existence of these local defect solutions.

Wednesday April 6

Marcus Webb, (University of Cambridge)

Computing Spectral Data of Infinite Jacobi Matrices

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Jacobi matrices are tridiagonal matrices with positive off-diagonal elements. Their interest for applied mathematics depends on the interpretation, such as, discretisations of differential operators, multiplication operators for series expansions in special functions, or through their connections to random matrix theory. In all cases, the spectrum of the operator is of paramount importance. In this talk I will discuss computing the spectrum via connection coefficients of orthogonal polynomials and via infinite dimensional QL iterations. As an example application, I will demonstrate a method for solving discrete Schodinger equations with finitely supported potentials which is faster (and arguably more natural) than a truncation approach. This is joint work with Sheehan Olver (USyd).

Wednesday January 13

Stephen Garrett (University of Leicester)

The stability of the boundary-layer flow over rotating spheres and cones

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The stability properties of the boundary layer over the rotating disk has been an active topic in the literature since the 1950s. Despite being initially motivated by the similarity with flows over swept wings, the proliferation of commercial wing technology and, more recently, the use of CFD in engineering design meant that the disk soon became a curiosity of fundamental science; the link between the real world and the rotating disk appeared to be broken. However, in the early 2000s it was realised that the flows over rotating mechanical components and ballistics (often simply cones and spheres) could be related directly to the flows over the disk. This re-established a link between the now extensive fundamental literature on the disk and real-world engineering design. In this talk I will explore the connections between the flows over rotating disks, cones and spheres, including the limits of the application of the disk literature to these other geometries. We will discover that, despite having more a definite value in engineering design, these related geometries are proving to be yet further sources of fundamental science in themselves. The talk will end with a summary of an ongoing analytical study of the rotating-sphere flow.
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