Nonlinear generalizations of the Schrödinger equation are of wide applicability to a range of areas including atomic and optical systems,
plasma physics and water waves. In this talk we revisit some principal excitations in atomic and optical systems (such as BoseEinstein condensates and photorefractive crystals), namely dark solitonic fronts in singlecomponent systems, and darkbright waves in multicomponent systems. Upon introducing them and explaining their existence and stability properties in one spatial dimension, we will extend them both in the form of stripes and in that rings in twodimensions, presenting an alternative (adiabaticinvariant based) formulation of their stability and excitations. We will explore their filamentary dynamics, as well as the states that emerge from their transverse (snaking) instability. Then, we will consider these structures even in three dimensions, in the form of planar, as well as spherical shell wave patterns and generalize our adiabatic invariant formulation there. Finally, time permitting, we will give some glimpses of how some of these dynamical features in 1d and 2d generalize in a multiorbital, timedependent quantum setting.
Past Industrial and Applied Mathematics Seminar
Further Information:
The lecture will take place in the Michael Dummett Lecture Theatre (Blue Boar quad, Christ Church).
The flow of a thin film down an inclined plane is an important physical phenomenon appearing in many industrial applications, such as coating (where it is desirable to maintain the fluid interface flat) or heat transfer (where a larger interfacial area is beneficial). These applications lead to the need of reliably manipulating the flow in order to obtain a desired interfacial shape. The interface of such thin films can be described by a number of models, each of them exhibiting instabilities for certain parameter regimes. In this talk, I will propose a feedback control methodology based on samefluid blowing and suction. I use the Kuramoto–Sivashinsky (KS) equation to model interface perturbations and to derive the controls. I will show that one can use a finite number of pointactuated controls based on observations of the interface to stabilise both the flat solution and any chosen nontrivial solution of the KS equation. Furthermore, I will investigate the robustness of the designed controls to uncertain observations and parameter values, and study the effect of the controls across a hierarchy of models for the interface, which include the KS equation, (nonlinear) longwave models and the full Navier–Stokes equations.
The NavierStokes paradigm does not capture thermal fluctuations that drive familiar effects such as Brownian motion and are seen to be key to understanding counterintuitive phenomena in nanoscale interfacial flows. On the other hand, molecular simulations naturally account for these fluctuations but are limited to exceptionally short time scales. A framework that incorporates thermal noise is provided by fluctuating hydrodynamics, based on the socalled LandauLifshitzNavierStokes equations, and in this talk we shall exploit these equations to gain insight into nanoscale free surface flows. Particular attention will be given to flows with topological changes, such as the coalescence of drops, breakup of jets and rupture of thin liquid films for which both analytic linear stability results and numerical simulations will be presented and compared to the results of molecular dynamics.
We present analysis and computations of a nonlocal thin film model developed by Kalogirou et al (2016) for a perturbed twolayer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasisolution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2πν−1/2 of the disturbance, is determined analytically. We also present recent results on time periodic solutions arising from HoofBifurcation of the primary solution branch.
(This work is in collaboration with D. Papageorgiou & E. Oliveira )
The problem of a bubble moving steadily in a HeleShaw cell goes back to Taylor and Saffman in 1959. It is analogous to the wellknown selection problem for SaffmanTaylor fingers in a HeleShaw channel. We apply techniques in exponential asymptotics to study the bubble problem in the limit of vanishing surface tension, confirming previous numerical results, including a previously predicted surface tension scaling law. Our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane.
In biological cells, genomic DNA is complexed with proteins, forming socalled chromatin structure, and packed into the nucleus. Not only the nucleotide (A, T, G, C) sequence of DNA but also the 3D structure affects the genomic function. For example, certain regions of DNA are tightly packed with proteins (heterochromatin), which inhibits expression of genes coded there. The structure sometimes changes drastically depending on the state (e.g. cell cycle or developmental stage) of the cell. Hence, the structural dynamics of chromatin is now attracting attention in cell biology and medicine. However, it is difficult to experimentally observe the motion of the entire structure in detail. To combine and interpret data from different modes of observation (such as live imaging and electron micrograph) and predict the behavior, structural models of chromatin are needed. Although we can use molecular dynamics simulation at a microscopic level (~ kilo basepairs) and for a short time (~ microseconds), we cannot reproduce longterm behavior of the entire nucleus. Mesoscopic models are wanted for that purpose, however hard to develop (there are fundamental difficulties).
In this seminar, I will introduce our recent theoretical/computational studies of chromatin structure, either microscopic (molecular dynamics of DNA or single nucleosomes) or abstract (polymer models and reactiondiffusion processes), toward development of such a mesoscopic model including local "states" of DNA and binding proteins.
References:
T. Kameda, A. Awazu, Y. Togashi, "Histone Tail Dynamics in Partially Disassembled Nucleosomes During Chromatin Remodeling", Front. Mol. Biosci., in press (2019).
Y. Togashi, "Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings", J. Phys. Chem. B 123, 14811490 (2019).
E. Rolls, Y. Togashi, R. Erban, "Varying the Resolution of the Rouse Model on Temporal and Spatial Scales: Application to Multiscale Modelling of DNA Dynamics", Multiscale Model. Simul. 15, 16721693 (2017).
S. Shinkai, T. Nozaki, K. Maeshima, Y. Togashi, "Dynamic Nucleosome Movement Provides Structural Information of Topological Chromatin Domains in Living Human Cells", PLoS Comput. Biol. 12, e1005136 (2016).

The dynamics of many physical systems often evolve to asymptotic states that exhibit periodic spatial and temporal variations in their properties such as density, temperature, etc. Such regular patterns look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range spatial order but no translational symmetry.
Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic, star, and block copolymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram.
By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such metastable localized quasicrystals is significant as they may affect the dynamics of the crystallisation in soft matter.
The talk will begin with an introduction to the science of what determines the behaviour of a liquid on a on a surface and giving an overview of some of the different theories that can be used to describe the shape and structure of the liquid in the drop. These include microscopic density functional theory (DFT), which describes the liquid structure on the scale of the individual liquid molecules, and mesoscopic thin film equation (PDE) and kinetic MonteCarlo models. A DFT based method for calculating the binding potential 𝑔(h) for a film of liquid on a solid surface, where h is the thickness of the liquid film, will be presented. The form of 𝑔(h) determines whether or not the liquid wets the surface. Calculating drop profiles using both DFT and also from inputting 𝑔(h) into the mesoscopic theory and comparing quantities such as the contact angle and the shape of the drops, we find good agreement between the two methods, validating the coarsegraining. The talk will conclude with a discussion of some recent work on modelling evaporating drops with applications to inkjet printing.