On this page is a collection of all past and future scheduled talks and their abstracts by session. Links to recorded versions of these talks will be provided where available.
The Boussinesq equations are a model commonly used to describe the propagation of weakly nonlinear waves in shallow water; they can be viewed as a dispersive perturbation of the (hyperbolic) nonlinear shallow water equation. The goal of this talk is to study the interactions of such waves with a freely floating objects; several models will be derived and analyzed mathematically. (Recording here)
We consider a nonlocal formulation of the water-wave problem for a free surface with an irrotational flow, and show how the problem can be reduced to a single equation for the interface. The formulation is also extended to constant vorticity and interfacial flows of different density fluids. We show how this formulation can be used to systematically derive Olver’s conservation laws not only for an irrotational fluid, but for constant vorticity and interfaces. This framework easily lends itself to computing the related conservation laws for various asymptotic models. (Recording here)
The Whitham equation was proposed as a generalization of the KdV equation that has the same linear dispersion relation as the unidirectional water wave problem. Recently, bidirectional Whitham systems, that is generalizations of the Whitham equation that have the same linear dispersion relation as the bidirectional water wave problem, have gained significant interest. We discuss three of these generalizations and compare their predictions with measurements obtained from experiments with flat and non-flat bathymetry. (Recording here)
The soliton structure plays a fundamental role in many physical systems due to its fundamental feature: its shape remains unchanged after the collision with another soliton in the case of integrable dynamics. Such particle-like behaviour has been at the origin of a new mathematical object: the soliton gas, consisting of an incoherent collection of solitons for which phases (positions) and spectral parameters (e.g. amplitudes) are randomly distributed. The study of soliton gases involves the description of the gas dynamics as well as the corresponding modulation of the nonlinear wave field statistics, which makes the soliton gas a particularly interesting embodiment of the particle-wave duality of solitons.
Motivated by the recent realisation of bidirectional soliton gases in a shallow water experiment, we investigate two integrable models of bidirectional wave: the nonlinear Schrödinger equation and the Kaup-Boussinesq equation. Using a physical approach, we derive the so-called kinetic equation that governs the gas dynamics for the two integrable systems. We notably show that the structure of the kinetic equation depends on the "isotropic" or the "anisotropic" nature of the solitons interaction. Additionally we derive expressions for statistical moments of the physical fields (e.g. mean water level). As an illustration of the theory, we solve numerically the gas shock tube problem describing the collision of two "cold" soliton gases. An excellent agreement with exact solutions of the kinetic equations is observed. (Recording here)
Potential-flow water waves are coupled to a shallow-water model, including hydraulic bores, at a beach to deal with unidirectional wave propagation in a finite domain. Such a set-up also matches wavetank conditions used for testing and validation of model structures such as ships in maritime engineering. Note that shorter, non-breaking and dispersive waves in deeper water are thus damped by wave breaking at the beach. A suitable coupling point is chosen in sufficiently shallow water, where the two models are coupled by using variational techniques. Numerically, a space-time variational approach is followed for the potential-flow water waves coupled to a classical finite-volume method for the shallow-water model. The entire approach has been validated numerically against bespoke wavetank experiments undertaken at the TU Delft. The main work was performed by Floriane Gidel [1], in collaboration with Tim Bunnik and Geert Kapsenberg (MARIN, Maritime Research Institute Netherlands), Mark Kelmanson and the speaker (Leeds). (Recording here)
[1] F. Gidel 2018: Variational water-wave models and pyramidal freak waves. PhD thesis. University of Leeds: http://etheses.whiterose.ac.uk/21730/
[2] O. Bokhove 2021: Variational water-wave modeling: from deep water to beaches. Book chapter. Mathematics of Marine Modeling. Springer. Eds. Deleersnijder, Heemink and Schuttelaars.
We study the irreversible dynamics of vortex reconnections in quantum fluids within the framework of the Gross–Pitaevskii model, a nonlinear Schroedinger-type equation. We quantitatively explain the time-asymmetry characterising the reconnection process by relating it to the emission of localised directional sound pulse. Our theoretical results shed new light on energy transfer and turbulence in fluid mechanics and have the prospect of being tested in quantum fluid experiments. (Recording here)
In the early fifties in Los Alamos E. Fermi in collaboration with J. Pasta, S. Ulam and M. Tsingou investigated a one dimensional chain of equal masses connected by a weakly nonlinear spring. The key question was related to the understanding of the phenomenon of conduction in solids; in particular they wanted to estimate the time needed to reach a statistical equilibrium state characterized by the equipartition of energy among the Fourier modes. They approached the problem numerically using the MANIAC I computer; however, the system did not thermailize and they observed a recurrence to the initial state (this is known as the FPUT-recurrence). This unexpected result has led to the development of the modern nonlinear physics (discovery of solitons and integrability). In this seminar, I will give an historical overview of the subject and present new results on the problem of thermalization based on the Wave Turbulence Theory. (Recording here)
In this talk, we will show how to reduce the problem of analysing stability of solutions to nonlinear Hamiltonian PDEs, to that of finding roots of polynomials. Using the Kawahara equation as an example, it will be shown how to obtain explicit expressions for regions of stability for different parameters in the equation. Finally, we will illustrate how this can approach can easily be extended to more general PDEs. (Recording here)
In the 1960’s G. B. Whitham suggested a non-local version of the KdV equation as a model for water waves. Unlike the KdV equation it is not integrable, but it has certain other advantages. In particular, it has the same dispersion relation as the full water wave problem and it allows for wave breaking. The existence of a highest, cusped periodic wave was recently proved using global bifurcation theory. I will discuss the same problem for solitary waves. This presents several new challenges. (Recording here)
Joint work with T. Truong (Lund) and M. Wheeler (Bath).
We use a simple yet Earth-like atmospheric model to propose a new framework for understanding the mathematics of blocking events, which are associated with low frequency, large scale waves in the atmosphere. Analysing error growth rates along a very long model trajectory, we show that blockings are associated with conditions of anomalously high instability of the atmosphere. Additionally, the lifetime of a blocking is positively correlated with the intensity of such an anomaly, against intuition. In the case of Atlantic blockings, predictability is especially reduced at the onset and decay of the blocking, while a relative increase of predictability is found in the mature phase, while the opposite holds for Pacific blockings, for which predictability is lowest in the mature phase. We associate blockings to a specific class of unstable periodic orbits (UPOs), natural modes of variability that cover the attractor of the system. The UPOs differ substantially in terms of instability, which explains the diversity of the atmosphere in terms predictability. The UPOs associated to blockings are indeed anomalously unstable, which leads to them being rarely visited. The onset of a blocking takes place when the trajectory of the system hops into the neighbourhood of one of these special UPOs. The decay takes place when the trajectory hops back to the neighbourhood of usual, less unstable UPOs associated with zonal flow. This justifies the classical Markov chains-based analysis of transitions between weather regimes. The existence of UPOs differing in the dimensionality of their unstable manifold indicates a very strong violation of hyperbolicity in the model, which leads to a lack of structural stability. We propose that this is could be a generic feature of atmospheric models and might be a fundamental cause behind difficulties in representing blockings for the current climate and uncertainties in predicting how their statistics will change as a result of climate change. (Recording here)
Reference
Lucarini, V., Gritsun, A. A new mathematical framework for atmospheric blocking events. Clim Dyn 54, 575–598 (2020). https://doi.org/10.1007/s00382-019-05018-2
In this talk we present recent results on the time evolution of broad banded, random inhomogeneous fields of deep water waves. Our study is based on solutions of the equation derived by Crawford, Saffman and Yuen in 1980. (“Evolution of a random inhomogeneous fields of nonlinear deep-water gravity waves”, Wave Motion, 2(1), 1 - 16).
This talk covers three major aspects. The instability of a homogeneous spectrum to inhomogeneous disturbances, the long time evolution of such instabilities, and their impact on the probability of encountering freak waves.
The main results are the following. First, it is shown, from a broad banded model, that a JONSWAP spectrum is unstable when it is narrow and it stabilizes as it broadens. Then, from the spectral time evolution, we obtained the evolution of the variance of the free surface. In case of instability, it is observed that the variance and thus, the energy in the wave field, localizes in regions of space and time. Initially, the evolution of the variance exhibits a recurrent pattern, akin to the one found in solutions of Alber’s equation. Interestingly, in the most unstable cases, such recurrent pattern fades away giving rise to a localized pattern that dominates the later stages of the evolution. Last, we compute the probabilities of encountering freak waves. Our results suggest significantly higher probabilities than those predicted from the Rayleigh distribution surpassing those obtained from Alber’s equation. (Recording here)
When following numerically branches of nonlinear water waves, one ends up ultimately with a 'solution' characterised by some kind of singularity. We refer to these 'singular solutions' as limiting configurations. The type of singularity observed depends on which effects are included in the model. For example the singularities are different for gravity waves, capillary waves, waves with vorticity and interfacial waves. We will review systematically these various singularities and present some new results which connect the various types of limiting configurations. (Recording here)
Ferrofluids, magnetic fluids consisting of iron nanoparticles, provide a good experimental medium to investigate properties of nonlinear waves and coherent structures. For a vertically applied magnetic field, there exists a critical field strength at which a surface instability occurs and spikes emerge from the ferrofluid, arranging in domain-covering cellular patterns. In 2005, solitary spikes were experimentally observed; these spikes were not affected by the shape of the fluid's container, and drifted around the domain, indicating they were localised solutions.
In this talk, I will introduce the ferrohydrostatic problem, formulated as a free-surface problem, and present our formal results for showing the existence of localised radial solutions via local invariant-manifold theory. This includes the introduction of an appropriate "spectral" decomposition, in order to reduce the problem to infinitely-many ODEs, and employing geometric 'de-singularisation' to identify exponentially-decaying solutions.
In particular, we show the existence of four classes of localised solutions to the ferrohydrostatic problem, and explore the parameter regions in which these localised radial patterns emerge. (Recording here)
In the evolution of nonlinear waves, localised structures and defects can form and persist, even within stable waves. One way that their formation can be understood is by using the Whitham Modulation equations (WMEs), a dispersionless set of quasilinear PDEs. However, a persistent problem is how to regularise this system via the inclusion of dispersive effects to prevent the emergence of multivalued wave quantities. Surprisingly, it transpires that such features already lurk within the WMEs whenever they are hyperbolic – one merely waits long enough in a suitable moving frame. This takes the form of the Korteweg – de Vries (KdV) equation, and is universal in the sense that its coefficients are tied to abstract properties of the original Lagrangian.
This leads to a more general question – can the properties of the characteristics be used to infer the resulting dynamics? This talk confirms this, and the connection between established concepts in hyperbolic systems (such as the Hamiltonian-Hopf bifurcation and linear degeneracy) and some well-known nonlinear dispersive equations, such as the Two-Way Boussinesq, modified KdV equations and fifth order KdV are made. (Recording here)
For most applications, water is assumed to be incompressible, irrotational and inviscid. Usually, these assumptions are enough to describe the main part of the dynamics of real water waves.
When viscosity is taken into account, vorticity also plays a role. Since the classical works of Lamb and Boussinesq in the XIX century, it is well-known that, under certain conditions, vorticity is important only in a layer near the free surface. With this in mind, Dias, Dyachenko & Zakharov proposed a free boundary problem modelling water waves with viscosity.
In this talk I will present new asymptotic model for water waves with viscosity together with new mathematical results for both the full free boundary problem proposed by Dias, Dyachenko & Zakharov and also for the asymptotic model of water waves. (Recording here)
The reversibility of light propagation in nonlinear media has recently received experimental attention, as it may lead to improvements in holography and imaging. This leads to a fundamental question – is the governing Nonlinear Schrodinger equation (NLS) reversible? We demonstrate that even though the NLS has a time-reversal symmetry, reversibility can become highly unlikely under noise or perturbations. Loss of reversibility in the NLS and other dynamical systems reveals the emergence of a preferred "arrow of time”, reminiscent of the thermodynamical one.
This “arrow of time” is also revealed by loss of phase – we show that as an ensemble of NLS solutions propagates, its phase information is lost in a statistical sense. An implication of this phenomena is that interactions between laser beams become chaotic and impossible to predict, contrary to the widespread belief. Not all is lost, however. Even though each interaction is unpredictable, the statistics of many interactions are predictable and follow a universal model. (Recording here)
Couder and Fort (PRL‘06) discovered that a fluid droplet bouncing on the surface of a vertically vibrating bath, forms a wave-particle system referred to as a hydrodynamic pilot-wave system. Such an object was only imagined in the quantum realm. Much research has been done since this discovery. Many mathematical problems arose, as will be outlined, where uncertainty related issues are recurrent. The main result of this talk regards a recent publication (N., Chaos, Sept.’18) where we show that two oscillating droplets, confined to separate wells, exhibit correlated features even when separated by a large distance. The particles’ phase space dynamics is described in a holistic fashion and may not be decomposed into separate subsystems. We detect “coherence” when the bouncing droplets behave as nonlinearly-coupled oscillators which synchronize spontaneously, as in the celebrated Kuramoto model for phase oscillators. The droplet coupling is dynamic and implicit, being wave-mediated as opposed to the Kuramoto model where phase-coupling is explicit and pre-defined. We also discover a regime where “coherence” is defined in a statistical manner.
In this talk, I will present an approximate first-order hyperbolic model for the hydrodynamic form of the defocusing nonlinear Schrödinger equation (NLS). This Euler-Korteweg type system can be seen as a Euler-Lagrange equation to a Lagrangian submitted to a mass conservation constraint. Due to the presence of dispersive terms, such a Lagrangian depends explicitly on the gradient of density. The idea is to create a new dummy variable that accurately approximates the density via a penalty method. Then, we take its gradient as a new independent variable and apply Hamilton's principle.
I will explain the main ideas behind the method, how the resulting system is hyperbolic, and present some obtained numerical results for gray solitons and dispersive shockwaves.
The inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. More specifically, inverse source scattering problem arises in many areas of science. It has numerous applications to medical imaging and geophysics, acoustical and bio-medical industries, antenna synthesis, and mechanical and material science. In particular, inverse source problem seeks the radiating source which produces the measured wave field. This research aims to provide a technique for recovering the source function of the Helmholtz equation and some classical system of PDEs from boundary data at multiple wave numbers when the source is compactly supported in an arbitrary bounded C_2− boundary domain, establish uniqueness for the source from the Cauchy data on any open non empty part of the boundary for arbitrary positive K, and increasing stability when wave number K is getting large for a 2 and 3 dimensional general domain. Various studies showed that the uniqueness can be regained by taking multifrequency boundary measurement in a non-empty frequency interval (0, K). (Recording here)
We study the linear dynamics of spectrally stable T-periodic stationary solutions of the Lugiato-Lefever equation (LLE), a damped nonlinear Schr¨odinger equation with forcing that arises in nonlinear optics. It is known that such T-periodic solutions are nonlinearly stable to NT-periodic, i.e. subharmonic, perturbations for each N ∈ N with exponential decay rates of the form e −δN t . However, both the exponential rates of decay δN and the allowable size of initial perturbations tend to 0 as N → ∞ so that this result is non-uniform in N and is, in fact, empty in the limit N = ∞. The primary goal of this talk is to introduce a methodology, in the context of the LLE, by which a uniform stability result for subharmonic perturbations may be achieved, at least at the linear level. (Recording here)
The Whitham equation is a non-local, non-homogeneous and weakly dispersive model for shallow water waves. Like in the case of the Stokes wave for Euler, non-smooth traveling waves with greatest height between crest and trough have been shown to exist for this model. In this talk I will discuss the existence of a unique, even and periodic traveling wave of greatest height solution to the Whitham equation. This wave of extreme form is unique in the class of monotone solutions and, moreover, it is convex between consecutive cusps. The talk is based on a joint work with A. Enciso and J. Gómez-Serrano. (Recording here)
Scalar equations modeling nonlinear wave evolution typically take the form of a conservation law often modified by long wave dispersion. However, some features of the original problem may be lost when higher order dispersive effects are neglected. One may modify the model equation and include full linear dispersion by use of a conservation law or higher order long wave dispersion by including more differential terms. In this presentation, we derive the set of Whitham modulation equations for a general class of nonlinear, dispersive equations with general nonlinearity and full linear dispersion. The Whitham modulation equations describe the slow evolution of a nonlinear, periodic wave. Following the derivation of the Whitham modulation equations, we use the fifth order Korteweg-de Vries equation as an example to demonstrate the novel nonlinear phenomena that can be described by the modulation system when dispersive terms higher than third order are included. In this portion of the talk, we focus on traveling wave solutions that correspond to discontinuous shock solutions of the Whitham modulation equations. (Recording here)
Dispersive shock waves (DSWs), or sometimes known as undular bores in fluid mechanics, are nonlinear dispersive wave phenomena that get generated when physical quantities undergo rapid variations in media whose dispersion dominates viscosity. In this talk, we will present various resonant optical DSW regimes arising in a nonlocal nonlinear medium so-called defocusing nematic liquid crystal. These DSW regimes are generated from discontinuous initial conditions for the optical field and are resonant in that linear dispersive waves are in resonance with the DSW, resulting in a resonant radiation propagating ahead of the DSW itself. Previous studies have used the classical KdV equation and gas dynamic shock wave theory to treat nematic DSWs, but poor agreements with numerical solutions were found. Indeed, the standard DSW structure disappears and a Whitham shock emerges as the initial jumps become sufficiently large. Asymptotic theory, approximate methods or Whitham’s modulation theory are used to find solutions for these resonant nematic DSWs. The comparisons between theoretical and numerical solutions are found excellent in all nematic dispersive hydrodynamic regimes. This is a collaboration project with Noel F. Smyth. (Recording here)
We prove that all solitary gravity waves are supercritical, i.e. they travel faster than infinitesimal periodic waves. Previously, this basic fact was only known under additional sign conditions or smallness assumptions. While there are physical grounds for expecting subcritical solitary waves to be extremely rare, it seems impossible to turn these ideas into a rigorous proof. Instead, our argument hinges on a new function which is related to the flow force and has several surprising properties.
This is joint work with Vladimir Kozlov and Evgeniy Lokharu. (Recording here)
There is increasing evidence that much of the rhythmic activity seen in the brain is not synchronous, but rather organized into various types of waves such as plane waves and rotating waves. In many of these experiments, the quantities measured are the spatial phase of the oscillation. Thus, a natural approach is to study the dynamics of coupled phase oscillators. Since interactions in the brain are nonlocal, in this talk, I will describe some recent results on equations of the form:
d u(x,t)/dt = omega(x) + int_D K(x-y)H(u(y,t)-u(x,t)) dy
where u(x,t) is the local spatial phase, omega(x) is the intrinsic frequency at x, K(x) is a convolution kernel, and H(u) is a periodic interation function. The domain, D will be either a ring, a line interval, or an annulus. We first review the behavior on a ring when the frequency is constant. Next, we study the existence and stability of rotating waves on an annulus as the size of the hole changes. Finally, we consider the line interval when there is a frequency gradient. We apply several techniques including conversion to boundary value problems and singular perturbation. This work is joint with graduate student Yujie Ding. (Recording here)
Fully localised solitary waves are travelling-wave solutions of the three-dimensional gravity-capillary water wave problem which decay to zero in every horizontal spatial direction. Their existence for water of finite depth has recently been established, and in this talk I present an existence theory for water of infinite depth. The governing equations are reduced to a perturbation of the two-dimensional nonlinear Schrödinger equation, which admits a family of localised solutions. Two of these solutions are symmetric in both horizontal directions and an application of a suitable version of the implicit-function theorem shows that they persist under perturbations.
This is joint work with Boris Buffoni (EPFL) and Erik Wahlén (Lund)
We consider a strongly nonlinear long wave model for large amplitude internal waves in a three-layer fluid bounded above and below by rigid boundaries. The model extends the two-layer Miyata-Choi-Camassa (MCC) model (Miyata 1988; Choi & Camassa 1999) and is able to describe the propagation of long internal waves of both the first and second baroclinic modes. Solitary-wave solutions of the model are shown to be governed by a Hamiltonian system with two degrees of freedom. Emphasis is given to the solitary waves of the second baroclinic mode (mode-2) and their strongly nonlinear characteristics that fail to be captured by weakly nonlinear models. We will show that in addition to the classical single-hump profiles predicted by the KdV weakly nonlinear theory, new classes of mode-2 solutions characterised by multi-humped wave profiles of large amplitude will be revealed in the case when the density transition layer is thin. Our analytical predictions based on asymptotic theory are then corroborated by a numerical study of the full dynamical system.
This is joint work with Wooyoung Choi (NJIT, USA) and Paul Milewski (Univ. Bath, UK)
Solitons are localised solutions of integrable systems that can be viewed as “particles” of complex statistical objects called soliton gases. In view of the growing evidence of their ubiquity in fluids and nonlinear optical media, such “integrable” gases are of fundamental interest for nonlinear physics. In particular, it has been shown recently (PRL 123, 234102 (2019)) that the long-time asymptotic behaviour of the noise-induced, spontaneous modulational instability can be accurately modelled by certain dense soliton gas dynamics. In my talk, I will outline nonlinear spectral theory of soliton gases based on a special, thermodynamic-type limit of multiphase (finite-gap) solutions and their modulations for the focusing nonlinear Schrödinger equation (PRE 101, 052207 (2020), joint with A. Tovbis). Wherever possible, connections with physical experiments will be highlighted.