## Spring 2017 Seminars and Abstracts Spring 2017 Seminars and Abstracts

### Hydrodynamic Interactions between Flagellar Filaments

Date: Monday 16th January, 2pm, (Queens 1.04)
Speaker: Dr Eric Lauga (University of Cambridge)
Abstract: Many small organisms possess flagella, slender whiplike appendages which are actuated in a periodic fashion in fluids and allow the cells to self-propel. In particular, most motile bacteria are equipped with multiple helical rotating flagella which interact hydrodynamically, synchronise, and can form a tight helical bundle behind a swimming cell. We consider here the problem of bundling and unbundling of these flagellar filaments. Most past theoretical work has approached the problem of bundling using numerical computations. Here, we present an asymptotic treatment of the interactions between elastic rotating filaments. We first show how to asymptotically compute the hydrodynamic kernels governing hydrodynamic interactions in the case of long filaments, and we then use these results to derive the nonlocal, nonlinear, equations of motions of each filament. We finally apply our results to a few simple configurations.

### Electron-MHD Evolution in Strongly Magnetised Neutron Stars

Date: Monday 23rd January, 2pm, (Queens 2.22)
Speaker: Dr Kostas Gourgouliatos (University of Leeds)
Abstract: Neutron stars host the strongest magnetic fields in the Universe that we know of, reaching 10^15 G in magnetars. However the origin of these magnetic fields and the way they power magnetar bursting activity is still under debate. In this seminar, I will present the Electron Magnetohydrodynamic formulation for the evolution of the magnetic field in neutron star crusts. I will discuss the results of plane parallel, axially symmetric and finally 3-D simulations, demonstrating that the combined effect of instabilities and secular evolution leads to the formation of very strong localised magnetic fields. These magnetic fields facilitate the observed bursts and the quiescent emission of magnetars. In particular instabilities can create pockets where the magnetic field is at least one order of magnitude stronger than the average magnetic field leading to a more economical magnetar theory.

### Invitation to Random Tensors

Date: Wednesday 25th January, 4pm, (SCI 3.05)
Speaker: Dr Razvan Gurau (Ecole Polytechnique, Palaiseau, France)
Abstract: Random matrices are ubiquitous in modern theoretical physics and provide insights on a wealth of phenomena, from the spectra of heavy nuclei to the theory of strong interactions or random two dimensional surfaces. The backbone of all the analytical results in matrix models is their 1/N expansion (where N is the size of the matrix). Despite early attempts in the '90, the generalization of this 1/N expansion to higher dimensional random tensor models has proven very challenging. This changed with the discovery of the 1/N expansion (originally for colored and subsequently for arbitrary invariant) tensor models in 2010. I this talk I will present a short introduction to the modern theory of random tensors and its connections to random higher dimensional geometry.

### Vortex Reconnections and Rebounds in Trapped Atomic Bose-Einstein Condensates

Date: Monday 30th January, 2pm, (JSC 2.02)
Speaker: Dr Luca Galantucci (Newcastle University)
Abstract: Reconnections and interactions of filamentary coherent structures play a fundamental role in the dynamics of classical and quantum fluids, plasmas and nematic liquid crystals. In quantum fluids vorticity is concentrated into discrete (quantised) vortex lines (unlike ordinary fluids where vorticity is a continuous field), turning vortex reconnections into isolated events, conceptually easier to study.
In order to investigate the impact of non-homogeneous density fields on the dynamics of quantum reconnections, we perform a numerical study of two-vortex interactions in magnetically trapped elongated Bose--Einstein condensates in the T=0 limit. We observe different vortex interactions regimes depending on the vortex orientations and their relative velocity: unperturbed orbiting, bounce dynamics, single and double reconnection events [1]. The key ingredients driving the dynamics are the anti-parallel preferred alignment of the vortices and the impact of density gradients arising from the inhomogeneity of the trapping potential.
The results are confirmed by ongoing experiments in Trento performed employing an innovative non--destrutive real--time imaging technique capable of determining the temporal evolution of the axial dynamics [2] and the orientation of the vortices.

[1] S. Serafini, L. Galantucci et al. , arXiv: 1611.01691 (2016)

[2] S. Serafini, M. Barbiero et al., Phys. Rev. Lett., 115, 170402 (2015)

### Experimental and Numerical Study of the Sloshing Motion in a Rectangular Tank with Multiple Vertical Cylinders

Date: Wednesday 1st February, 3pm, (SCI 1.20)
Speaker: Prof Bernard Molin (École Centrale de Marseille, France)
Abstract: Sloshing tests are performed on a rectangular tank filled with bottom-mounted vertical cylinders, around the natural frequency of the first sloshing mode. The numbers and heights of the cylinders are varied. From the position of the resonance peak an experimental dispersion equation is derived, and compared with different formulations from literature. A new formulation is proposed. A modal approach is then applied to derive the RAO of the sloshing response, and compared with the experimental results.

### Time-Dependent Conformal Mappings with Applications to Nonlinear Sloshing Problems

Date: Monday 6th February, 2pm, (Queens 2.22)
Speaker: Dr Matthew Turner (University of Surrey)
Abstract: In this talk we examine two key features of time-dependent conformal mappings in doubly-connected regions, the evolution of the conformal modulus, Q(t), and the boundary transformation generalizing the Hilbert transform. Results of this theory are applied to inviscid, incompressible, irrotational fluid sloshing in a rectangular vessel. It is shown that the explicit calculation of the conformal modulus is essential to correctly predict features of the flow, such as the free surface evolution.
We also present results for fully dynamic simulations which use a time-dependent conformal mapping and the Garrick generalization of the Hilbert transform to map the physical domain to a time-dependent rectangle in the computational domain. The results of this new approach are compared to the complementary numerical scheme of Frandsen (2004)  and it is shown that correct calculation of the conformal modulus is essential in order to obtain agreement between the two methods.

### A New Limiter for 2D & 3D Mixed Modal Discontinues Galerkin to Solve Compressible Navier-Stokes Equations

Date: Monday 13th February, 2pm, (JSC 2.02)
Speaker: Dr Hong Xiao (Northwestern Polytechnical University, Xian, China, and University of Cambridge)
Abstract: High-order Discontinuous Galerkin (DG) methods have shown a lot of promise in being able to provide high accuracy and efficiency as well as the flexibility to handle complex unstructured grids. It is well known that the DG method combines advantageous features of both finite volume (FVM) and finite element (FEM) methods. It obtains high accuracy by means of higher-order polynomial approximation within an element. However, similar to any higher resolution schemes, DG methods are suffering from non-physical oscillations in the vicinity of discontinuities. Generally, the limiters are used to reduce the order of accuracy in the presence of discontinuity. Most of the approached that have been proposed are for Nodal DG. In Nodal DG, the shape functions are of the nodal form like Lagrange interpolation that once they constructed for a certain set of points over the element interval then can not be re-evaluated at any other location inside the element. However in Modal DG, the degrees of freedom are the modal shape functions and they can be reconstructed at any location over the element. Although several limiters have been developed for Nodal DG, the limiters for Modal DG are not well understood because of higher freedom in modal functions reconstruction. A new high order limiter for Modal DG will be discussed in this presentation.  Also, the presentation will provide the detailed implementation and validation of the new limiter based DG method including shock/vortex interactions, the gas flow around the 2D airfoil, the subsonic and hypersonic gas flow around 2D cylinder and 3D sphere, the hypersonic gas flow around 3D complex vehicle.

### Using pulsars in a Galactic-scale Gravitational-Wave Detector

Date: Monday 20th February, 2pm, (SCI 1.20)
Speaker: Dr Robert Ferdman (UEA)
Abstract: At the forefront of observation astrophysics has been the direct detection of gravitational waves.  In particular, the recent discoveries by the LIGO/Virgo collaboration has not only validated a key prediction of Einstein's general theory of relativity, but has heralded the era of gravitational-wave astrophysics.   In this talk, I will focus on the international effort to use the distances between Earth and several millisecond-period pulsars -- the compact remnants of massive starts that have exploded in supernova events -- as arms of a Galactic-scale gravitational-wave detector. This "Pulsar Timing Array" is sensitive to the nanohertz frequency region of the gravitational-wave spectrum, and is the only current method for studying the emission from supermassive black hole pairs. It is therefore complementary to the larger frequency ranges probed by ground-based detectors, which will be sensitive to sources such as merging neutron-star or stellar-mass black hole binaries. I will discuss how we will be able to detect gravitational waves with pulsar timing, and describe recent progress.  I will also briefly describe ongoing and future instrumentation and other observational considerations that will greatly benefit this work.

### Fluids of Light: From Superflidity to Analogues for Gravitational Phenomena

Date: Monday 6th March, 2pm, (JSC 1.02)
Speaker: Prof. Daniele Faccio (Heriot-Watt University)
Abstract: Recent years have seen a renewed interest for so called “fluids of light” or “photon fluids”. Since the first works by Chiao in the ‘90s, a fluid-like behaviour of light has been found in many different systems, ranging from exciton-polaritons to lasers. This fluid-like behaviour ultimately lies in the fact a repulsive photon-photon interaction, mediated by a material nonlinearity, leads to a propagation equation for light that can be recast in the form of a contintuity and an Euler equation. Over the past few years we have been investigating a relatively simple system: a laser beam passes through a slightly absorbing thermo-optic material whose refractive index decreases with increasing light power. The resulting repulsive photon-photon interaction is shown to lead to a superfluid behaviour: we have measured the superfluid dispersion relation and we have observed the spontaneous break up into quantised vortices when the fluid flows around an extended obstacle. However, this fluid also exhibits some unique features due to the nonlocal nature of the thermal medium. This nonlocality can be embraced to study novel effects such as collective shocks and to experimentally study solutions to the Schrodinger-Newton equation. Finally, I will discuss how superfluids, and in particular our photon fluid can be used to study the kinematics of rotating black holes. We have identified for the first time an experimental system, a single vortex with a radial flow, where the horizon and ergosphere are physically separated due to the angular momentum of the vortex rotation. I will discuss ongoing work aimed at studying Penrose superradiance, i.e. amplification of scattered waves from the rotating black hole vortex.

### Joint Pure and Applied Seminar (PhD Students)

Date: Monday 13th March, 2pm, (SCI 1.20)
Speaker: Jack Keeler
Title: Forced Solitary Waves with Algebraic Decay at Critical Froude Number
Abstract: Flow over bottom topography at critical Froude number (when F=1 ) is examined with an emphasis on steady, forced solitary wave solutions with algebraic decay in the far-field. We focus on the case of a negative Gaussian topography representing a smooth bottom dip (steady solutions for critical flow are not possible for positive-definite topographies). We study the flow in the weakly-nonlinear limit by way of the forced KdV equation and use a combination of numerical and asymptotic methods to probe the steady solution space, and to conduct time-dependent simulations.
For large amplitude negative Gaussian forcing, we construct an asymptotic solution using boundary layer theory; one point of interest here is an internal layer away from the origin which mediates a change from exponential decay away from the central dip to algebraic decay in the far-field. Intriguingly solutions with different numbers of waves trapped around the central dip are also found for large amplitude topography but these cannot be captured by the boundary-layer analysis. In fact a seemingly infinite sequence of solution branches is uncovered using numerical methods, and in general the solution for any given topography amplitude is non-unique.

Speaker: Gabriele Bocca
Title: Mutation of Cyclic Quivers and Cutting Sets.
Abstract: Path algebras of quivers with relations are very important in Representation Theory since they provide an easy description of a wide class of finite dimensional algebras. For some particular classes of path algebras it is possible to define rules to "mutate" a quiver into one other such that the corresponding algebras share some useful algebraic properties. In my talk I will describe this procedure for a particular class of algebras that can be represented by path algebras over a cyclic quiver with relations.

### Shallow-Water Models for a Vibrating Fluid

Date: Monday 20th March, 2pm, (SCI 1.20)
Speaker: Dr Konstantin Ilin (University of York)
Abstract: We consider a layer of an inviscid fluid with free surface which is subject to vertical high-frequency vibrations. Three asymptotic systems of equations that describe slowly evolving (in comparison with the vibration frequency) free-surface waves will be discussed. The first set of equations is obtained without assuming that the waves are long. These equations are as difficult to solve as the exact equations for irrotational water waves in a non-vibrating fluid. The other two models describe long waves. These models are obtained under two different assumptions about the amplitude of the vibration. Surprisingly, the governing equations have exactly the same form in both cases (up to interpretation of some constants). These equations reduce to the standard dispersionless shallow-water equations if the vibration is absent, and the vibration manifests itself via an additional term which is similar to the term that would appear if surface tension were taken into account. We shall show that our dispersive shallow water equations have both solitary and periodic travelling waves solutions and discuss an analogy between these solutions and travelling capillary-gravity waves in a non-vibrating fluid. Some weakly-nonlinear models will also be considered.

### Joint Pure and Applied Seminar

Date: Monday 24th April, 2pm, (JSC 2.03)
Speaker: Dr Hayder Salman (UEA)
Title: From Differential Geometry to the Phase Field in Quantum Mechanics
Abstract: In this talk, I will discuss three disparate and seemingly unrelated areas to help understand the properties of the phase of a quantum wave-function; namely differential geometry, fluid mechanics, and quantum mechanics. In quantum mechanics, the state of a system is described by a complex scalar field $\psi:\mathbb{R}^3\times \mathbb{R} \rightarrow \mathbb{C}$, the so-called wave-function of a system. It follows that the phase of this complex scalar field determines the subsequent time evolution of the system even though, in practice, it can not be measured directly. It is now well established that although the time development of this complex scalar field is governed by a Schr\"{o}dinger equation, an alternative fluid-mechanical interpretation exits in which the modulus and argument of $\psi$ are related to the density and velocity potential of the fluid. In classical fluids, under suitable assumptions, the velocity potential is considered to be a useful construct from which the physically more relevant velocity field can be derived. However, its interpretation as the quantum-mechanical phase elevates it to a more fundamental status. The correspondence between the quantum-mechanical and the fluid-mechanical views allows phase defects to be identified with vortex line filaments embedded in 3D space. The possibility to knot these line filaments in both optical and superfluid systems, allowing knotted optical fields as well knotted superfluid vortices to be realized, have generated further interest in understanding properties of the quantum phase.
I will show how the phase can be understood using important results from differential geometry. Starting with properties of a planar curve, I will subsequently present generalisations of these results for a curve on a manifold that relates its local geometric properties to the global topological properties of the manifold. These results culminate in a generalized form of the Gauss-Bonnet theorem. An important ingredient in the formulation we present is an application of Elie Cartan's spinors for representing the group of SO(3) rotations of a bi-vector. By combining results from these different areas, we will show how a mathematically elegant expression can be obtained for the quantum mechanical phase/velocity potential that is analogous to the Biot-Savart law that arises in vortex dynamics and electromagnetism. The formulae we will present generalises previous results cited in classic texts on the subject.

### Total Positivity: A Concept at the Interface Between Algebra, Analysis and Combinatorics

UEA Mathematics Colloquium (Joint Pure and Applied Research Seminar)
Date: Monday 8th May 2pm, (LT3)
Speaker: Prof. Alan Sokal (University College London and New York University
Abstract: A matrix $M$ of real numbers is called {\em totally positive} if every minor of $M$ is nonnegative. This somewhat bizarre concept from linear algebra has surprising connections with analysis --- notably polynomials and entire functions with real zeros, and the classical moment problem and continued fractions --- as well as combinatorics. I will explain briefly some of these connections, and then introduce a generalization: a matrix $M$ of polynomials (in some set of indeterminates) will be called {\em coefficientwise totally positive} if every minor of $M$ is a polynomial with nonnegative coefficients.   Also, a sequence $(a_n)_{n \ge 0}$ of real numbers (or polynomials) will be called {\em (coefficientwise) Hankel-totally positive}\/ if the Hankel matrix $H = (a_{i+j})_{i,j \ge 0}$  associated to $(a_n)$ is (coefficientwise) totally positive. It turns out that many sequences of polynomials arising in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive; in some cases this can be proven using continued fractions, while in other cases it remains a conjecture.

Total Positivity Seminar Slide (PDF, 203 KB)

### Discontinuous Galerkin Methods and Supporting Computational Tools for Environmental Fluid Dynamics Modelling

Date: Tuesday 9th May, 1pm, (SCI 3.05)
Speaker: Dr Ethan Kubatko (The Ohio State University)
Abstract: Discontinuous Galerkin (DG) methods are a family of finite element methods that exhibit a number of favourable properties for modelling environmental fluid dynamics problems, including their ability to handle advection-dominated flow scenarios, their local conservation properties and the relative ease with which both $h$ (mesh) and $p$ (polynomial) refinement can be implemented. This talk will highlight the development and application of a suite of DG models for one-, two- and three-dimensional shallow water flow and overland flow due to rainfall. Supporting computational tools that are used within the context of these models include an advanced unstructured mesh generator that we have developed called AdMesh+ and new (in many cases optimal) sets of numerical integration rules and time stepping methods that have been specifically designed for efficient calculation when used with high-order DG spatial discretizations. A number of applications that demonstrate the accuracy, efficiency and robustness of the developed modeling framework will be highlighted.Discontinuous Galerkin (DG) methods are a family of finite element methods that exhibit a number of favourable properties for modelling environmental fluid dynamics problems, including their ability to handle advection-dominated flow scenarios, their local conservation properties and the relative ease with which both $h$ (mesh) and $p$ (polynomial) refinement can be implemented. This talk will highlight the development and application of a suite of DG models for one-, two- and three-dimensional shallow water flow and overland flow due to rainfall. Supporting computational tools that are used within the context of these models include an advanced unstructured mesh generator that we have developed called AdMesh+ and new (in many cases optimal) sets of numerical integration rules and time stepping methods that have been specifically designed for efficient calculation when used with high-order DG spatial discretizations. A number of applications that demonstrate the accuracy, efficiency and robustness of the developed modeling framework will be highlighted.

### Modelling & Computation of Capillary Microflows

Date: Monday 15th May, 2pm, (SCI 0.31)
Speaker: Dr James Sprittles (University of Warwick)
Abstract: Understanding the formation of liquid drops, their interaction with solid surfaces and their collisions with surrounding drops is the key to optimising a whole host of processes ranging from 3D printing to cloud formation. Accurate experimental observation of these phenomena is complex due to the small spatio-temporal scales or interest and, consequently, mathematical modelling and computational simulation become key tools with which to probe such flows.
Drop formation, dynamic wetting and coalescence are all so-called 'singular' capillary flows, in which classical modelling approaches lead to infinite values of flow variables and computation becomes increasingly complex. In this talk, I will describe the mathematical models proposed for this class of flows and the techniques which have been used to obtain both analytic and computational results. Simulations will then reveal the applicability of similarity solutions and new unexpected flow behaviours.
Finally, I will describe the development of models to describe gas microfilms that appear in both drop collisions and in dynamic wetting. These microfilms cannot be described by the Navier--Stokes equations and instead require the development of a model based on the kinetic theory of gases. Simulation results obtained using this model will be discussed and compared to experimental data that highlight the influence of the surrounding gas.

## Autumn 2016 Seminars and Abstracts Autumn 2016 Seminars and Abstracts

### Unsteady Free-Surface Flow Cased by a Moving Circular Cylinder

Date: Monday 24th October, 3pm, (SCI 3.05)
Speaker: Dr Vasil Kostikov (Novosibirsk State University, Russia)
Abstract: A problem on non-stationary free surface flow of an infinitely deep ideal fluid generated due to the motion of a submerged body is considered. The initial formulation of the problem is reduced to an integral-differential system of equations for the functions defining the free surface shape, the normal and tangential components of velocity on the free boundary. Small- time asymptotic solution is constructed for the case of circular cylinder that moves with a constant acceleration from rest. The role of non-linearity is clarified by the analysis of this solution in the context of formation mechanism of added mass layers, splash jets and finite amplitude surface waves

### Challenges for Climate and Weather Prediction in the Era of Exascale Computer Architectures: Oscillatory Stiffness, Time-Parallelism and the Role of Long-Time Dynamics

Date: Monday 7th November, 2pm, (LT4)
Speaker: Prof. Beth Wingate (University of Exeter)
Abstract: For weather or climate models to achieve exascale performance on next-generation heterogeneous computer architectures they will be required to exploit on the order of hundred-million-way parallelism. This degree of parallelism far exceeds anything possible in today's models even though they are highly optimized. In this talk I will discuss one of the mathematical issues that leads to the limitations in space- and time-parallelism for climate and weather prediction models oscillatory stiffness in the PDE. Many PDEs used in weather and climate simulations have the form: PDF File where the linear operator L has pure imaginary eigenvalues, the nonlinear term $N(u,u)$ is of polynomial type, the operator $D$ represents dissipation, and $\epsilon$ is a small non-dimensional parameter. The operator $\epsilon^{-1}L$ results in time oscillations on an order $O(\epsilon)$ time scale, and generally necessitates small time steps if standard explicit numerical integrators are used. Even implicit integrators need to use small time steps if accuracy is required. I will discuss the role of resonances in the PDEs in formulating new numerical algorithms with the potential to go beyond the strong- and weak-scaling limitations that presently exist.

### Modulation Instability of Finite Amplitude Periodic Travelling Waves: Theme and Variations

Date: Monday 14th November, 2pm, (LT4)
Speaker: Prof. Tom Bridges (University of Surrey)
Abstract: The most well known and well understood example of modulation instability is the Benjamin-Feir instability of weakly nonlinear Stokes waves.  On the other hand, modulation instability for finite-amplitude periodic travelling waves is much less understood, with the only nonlinear theory being Whitham modulation theory which is dispersionless and produces shocks or blowup.
However, at the transition from stability to instability more precise results can be obtained including nonlinearity and dispersion, as well as saturation or enhancement. The talk will first give an overview of the weakly nonlinear case, including the work of Johnson which derives a higher order NLS equation at the transition point.  Then a new theory will be presented for nonlinear behaviour near finite amplitude modulational instability transition. Examples and implications for the modulational transition of finite-amplitude Stokes waves, as well as simplified examples where the complete theory can be worked out in detail. This talk is based on joint work with Daniel Ratliff.

### Mean Flows in 2D Turbulence

Date: Monday 21st November, 2pm, (SCI 1.20)
Speaker: Dr Jason Laurie (Aston University)
Abstract: An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a large-scale condensate or mean flow--for a square aspect ratio this is a pair of coherent system-size vortices. We present a new theoretical analysis based on momentum and energy exchanges between the mean flow and the underlying turbulence and show that the mean velocity profile has an universal internal structure independent on the mechanisms of small-scale dissipation and small-scale forcing. We verify the theoretical predictions through extensive numerical simulations of the two-dimensional Navier-Stokes equations. We begin our analysis by investigating the square geometry before studying larger aspect ratios and predictions for zonal mean flows.

### Variational Modelling of Water Waves and Their Impact on Moving Ships

Date: Monday 28th November, 2pm, (EFRY 1.01)
Speaker: Dr Anna Kalogirou (University of Leeds)
Abstract: The study of water waves has been an important area of research for years; their significance becomes obvious when looking at ocean and offshore engineering or naval architecture. Local weather and sea conditions can often lead to extreme wave phenomena, e.g. waves with irregular height. Waves with anomalously high amplitudes relative to the ambient waves are called rogue waves and can appear either at the coast or in the open ocean. The aim of this study is to investigate mathematically the generation and interaction of such waves and their impact on wave-energy devices and moving ships. The modelling is demonstrated by analysing variational methods asymptotically and numerically.
A reduced potential flow water-wave model is derived, based on the assumptions of waves with small amplitude and large wavelength. This model consists of a set of modified Benney-Luke equations describing the deviation from the still water surface and the velocity potential at the bottom of the domain. A novel feature in our model is that the dynamics are non-autonomous due to the explicit dependence of the equations on time. Numerical results obtained using a (dis)continuous Galerkin finite element method (DGFEM) are compared to a soliton splash experiment in a long water channel with a contraction at its end, resulting after a sluice gate is removed at a finite time. The removal of the sluice gate is included in the variational principle through a time-dependent gravitational potential.
The Benney-Luke approximation for water waves is also adapted to accommodate nonlinear ship dynamics. The new model consists of the classical water-wave equations, coupled to a set of equations describing the dynamics of the ship. We will first investigate the dynamics of the coupled system linearised around a rest state. For simplicity, we also consider a simple ship structure consisting of V-shaped cross-sections. The model is solved numerically using a DGFEM and the numerical results are compared to observations from experiments in wave tanks that employ geometric wave amplification to create nonlinear rogue-wave effects.

### Shock Waves in Non-Convex Dispersive Hydrodynamics

Date: Monday 12th December, 2pm, (LT3)
Speaker: Dr Gennady El (Loughborough University)
Abstract: Dispersive shock waves (DSWs) are unsteady nonlinear wavetrains replacing traditional shocks in media whose dominant wavebreaking regularisation mechanism is dispersion. DSWs are as ubiquitous as solitons and have been observed in a broad range of physical media including water waves and optical fibres. When the governing system represents a convex hyperbolic conservation law modified by weak dispersion,  as in the KdV equation, the resulting DSW has a well-defined structure  that includes a modulated periodic wave train, led by a solitary wave. The mathematical description of such DSWs in the framework of the Whitham modulation theory  is now well understood [1].
The DSW structure is also seen in non-convex dispersive-hydrodynamic systems, the modified KdV (mKdV) equation being probably the simplest example. Non-convexity of the hyperbolic flux in mKdV leads  to a rich set of unsteady solutions not observed in the convex case. Despite the fundamental nature of the problem such solutions have been obtained only recently. They are of direct relevance to the propagation of internal undular bores in the ocean and to the nonlinear dynamics of polarisation waves in two-component Bose-Einstein condensates.  In my talk I will describe a complete set of solutions to the Riemann initial value problem for the mKdV equation and show their (nontrivial) connections with the known solutions of the mKdV-Burgers equation such as undercompressive shocks and shock-rarefactions [2].
If time permits, I will also describe  expansion shock solutions of the Benjamin-Bona-Mahony (BBM) equation, a new type of shocks owing their existence to the particular type of non-convex linear dispersion.  An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition [3].

[1] G.A. El and M.A. Hoefer, Dispersive shock waves and modulation theory, Physica D 33, 5 - 67 (2016).

[2] G.A. El, M.A. Hoefer and M. Shearer, Dispersive and diffusive-dispersive shocks for non-convex conservation laws, SIAM Review 59, (2017) in press.

[3] G.A. El, M.A. Hoefer and M. Shearer, Expansion shock waves in regularised shallow water theory, Proc. Roy. Soc.  A, 472, 20160141 (2016).