Speakers
Upcoming talks (2024-2025)
Past talks and recordings (please note, recordings not available for all speakers)
September 14th, 2020
Michael Weinstein (Department of Applied Physics and Applied Mathematics, and Department of Mathematics, Columbia University)
Title: Dynamics of waves in continuum honeycomb structures
Abstract: We overview results on the dynamics of waves (eg Schroedinger and Maxwell equations) in honeycomb structures and their deformations. We study phenomena which arise from the presence of Dirac (conical) points in the bulk band structure. These include the existence of robust edge (interface) states, which localize along certain sharp terminations, and along domain walls. We also discuss recent work on the emergence of pseudo-magnetic fields in non-uniformly deformed honeycomb structures. We apply these results to a predict Landau-like energy levels in photonic crystals, and present numerical confirmation of the theory.
September 28th, 2020
Bjorn Sandstede (Division of Applied Mathematics, Brown University)
Title: Dynamics of spiral waves
Abstract: Spiral waves are striking spatio-temporal structures that often organize the dynamics in dissipative, spatially extended systems. In this talk, I will first give an overview of existence and stability properties of spiral waves. Afterwards, I plan to discuss two recent results about spectral and linear stability of spiral waves. The first result indicates that planar spiral waves that exhibit transverse instabilities in the far field are spectrally stable but linearly unstable. The second result shows that spiral spectra are discontinuous in the zero-diffusion limit for reaction-diffusion models. These results are joint work with Stephanie Dodson and Arnd Scheel.
October 12th, 2020
Mark Ablowitz (Department of Applied Mathematics, University of Colorado at Boulder)
Title: Nonlinear Waves, Topological Insulators, Integrability
Abstract: Longitudinally driven periodic optical lattices, called `Floquet Topological Insulators' and properties will be introduced. They admit a class of linear and nonlinear edge waves that propagate unidirectionally without backscatter from defects along boundary edges. The envelope of the underlying discrete nonlinear edge wave satisfies the classical integrable Nonlinear Schrödinger equation. New classes of integrable nonlocal equations and how they can be derived from well-known physical systems will also be discussed.
October 26th, 2020
Thanasis Fokas (Department of Applied Mathematics and Theoretical Physics, Cambridge University)
Title: The Unified Transform: from the Wiener-Hopf technique to the solution of the x-periodic problem for nonlinear integrable PDEs
Abstract: For many years, the employment of the Wiener-Hopf technique to acoustics and other physical problems, was the only manifestation in applications of the Riemann-Hilbert formalism. However, in the last 50 years this formalism has appeared in a large number of problems in mathematics and mathematical physics. In particular, it played a crucial role for the development of a novel, hybrid numerical-analytic method for solving boundary value problems (Fokas Method, www.wikipedia.org/wiki/Fokas_method). Interesting, for linear PDEs, one of the most recent applications of this method is in problems where the traditional Wiener-Hopf technique is ineffective. For nonlinear PDEs, perhaps the most striking application of the new method is the solution of the x-periodic problem for integrable PDEs, obtained jointly with Bernard Deconick and Jonatan Lenells. This problem was first considered in the early 1970s; despite the involvement of many distinguished mathematician, the solution of the general initial value problem remained open and only particular solutions, the famous multi-gap solutions, were constructed. This lecture will review the above novel method, will briefly mention its successes for linear PDEs, and then will concentrate on the x-periodic problem for the NLS.
November 9th, 2020
Beatrice Pelloni (School of Mathematical and Computer Sciences, Heriot Watt University)
Title: The phenomenon of dispersive revivals
Abstract: I will give an introduction to the phenomenon of “dispersive quantisation” or “revivals”. Although first reported in 1835 by Talbot, this phenomenon was only studied for the periodic free space Schroedinger equation by Berry and al in the 1990s and rediscovered for the Airy equation by Peter Olver in 2010. A sizeable literature has examined revivals for the periodic problem for linear dispersive equations with polynomial dispersion relation. What I will discuss in this talk is further occurrences of this phenomenon for different boundary conditions, a novel form of revivals for more general dispersion relations and nonlocal equations such as the linearised Benjamin-Ono equation, and nonlinear (integrable) generalisations. This work is joint with Lyonell Boulton, Peter Olver and David Smith.
November 23rd, 2020
Dmitry Pelinovsky (Department of Mathematics and Statistics, McMaster University)
Title: Periodic travelling waves in nonlinear wave equations: modulation instability and rogue waves
Abstract: I will overview the following different wave phenomena in integrable nonlinear wave equations:
(1) universal patterns in the dynamics of fluxon condensates in the semi-classical limit;
(2) modulational instability of periodic travelling waves;
(3) dynamics of double-periodic waves in space and time;
(4) rogue waves on the background of periodic and double-periodic waves.
Main examples include the sine-Gordon equation, the nonlinear Schroedinger equation, and the derivative nonlinear Schroedinger equation. For the latter equation, in collaboration with Jinbing Chen (South East University, China) and Jeremy Upsal (University of Washington, USA), we adapted the method of nonlinearization of the Lax system in order to characterize the existence and modulation stability of periodic travelling waves. We give precise information on the location of Lax and stability spectra, with assistance of numerical package based on the so-called Hill's method. Particularly interesting outcome is the explicit relation between the existence of modulation instability and the existence of a rogue wave (localized solution in space and time) on the background of periodic travelling waves.
December 7th, 2020
Barbara Prinari (Department of Mathematics, University at Buffalo)
Title: Inverse scattering transform, solitons and rogue wave solutions of certain matrix nonlinear Schrodinger equations
Abstract: We will discuss the Inverse Scattering Transform (IST), solitons and soliton interactions, and rogue wave solutions for certain matrix nonlinear Schrodinger (NLS) equations. The class of equations includes the focusing and defocusing matrix NLS equations introduced by Wadati in 2004 in the context of spinor systems, and two reductions of the matrix NLS which are the analog of the modified Manakov system with mixed signs of the nonlinear coefficients, i.e., a nonlinearity in the norm which is of Minkowski type, instead of Euclidean type.
January 18th, 2021
Eugene Wayne (Mathematics Department, Boston University)
Title: Breathers as metastable states for weakly damped lattices of Hamiltonian oscillators
Abstract: We discuss the flow of energy in a lattice of Hamiltonian oscillators with weak damping at one end of the lattice. We derive bounds on the rate of dissipation when the initial energy in the lattice is localized in a spatially distant part of the lattice. For a special model, we exhibit a family of breather solutions for the undamped problem and show that the rate of energy dissipation can be explained by a very slow drift along this family of breathers. This is joint work with Noé Cuneo (Univ. of Paris 7), Jean-Pierre Eckmann (Univ. of Geneva) and Daniel Caballero (Boston Univ.)
February 1st, 2021
Anna Vainchtein (Department of Mathematics, University of Pittsburgh)
Title: Traveling waves in a driven Frenkel-Kontorova lattice
Abstract: Variants of Frenkel-Kontorova model, originally proposed to describe dislocations in crystal lattices, have been widely used to study a variety of physical phenomena, including dynamics of twin boundaries and domain walls, crystal growth, charge-density waves, Josephson junctions and DNA denaturation. In this talk I will discuss properties and stability of traveling waves in chains of Frenkel-Kontorova type driven by a constant external force. After reviewing some earlier studies for piecewise-smooth variants of the model, where exact and semi-analytical solutions can be constructed, I will describe numerical results for a fully nonlinear damped driven chain from a recent work with J. Cuevas-Maraver (U. of Sevilla), P. Kevrekidis (U. of Mass.) and H. Xu (Huazhong U.). In this setting, traveling wave solutions are computed as fixed points of a nonlinear map. We show that kinetic relation between the driving force and the velocity of the wave can become non-monotone at small velocities, due to resonances with linear modes, and also at large velocities, where it becomes multivalued. Exploring the spectral stability of the obtained waveforms, we identify, at the level of numerical accuracy of our computations, a precise criterion for instability of the traveling wave solutions: monotonically decreasing portions of the kinetic curve always bear an unstable eigendirection.
February 15th, 2021
Edgar Knobloch (Department of Physics, UC Berkeley)
Title: Geostrophic turbulence and the formation of large scale structure
Abstract: Low Rossby number convection is studied using an asymptotically reduced system of equations valid in the limit of strong rotation. The equations describe four regimes as the Rayleigh number $Ra$ increases: a disordered cellular regime near threshold, a regime of weakly interacting convective Taylor columns at larger $Ra$, followed for yet larger $Ra$ by a breakdown of the convective Taylor columns into a disordered plume regime characterized by reduced heat transport efficiency, and finally by a new type of turbulence called geostrophic turbulence. Properties of this state will be described and illustrated using direct numerical simulations of the reduced equations. These simulations reveal that geostrophic turbulence is unstable to the formation of large scale barotropic vortices or jets, via a process known as spectral condensation. The details of this process will be quantified and its implications explored. The results are corroborated via direct numerical simulations of the the Navier-Stokes equations; in the presence of boundaries robust boundary zonal flows resembling topologically protected edge states in chiral systems are present.
March 1st, 2021
Catherine Sulem (Department of Mathematics, University of Toronto)
Title: Normal form transformations and Dysthe's equation for the nonlinear modulation of deep-water gravity waves
Abstract: I will present a new Hamiltonian version of Dysthe's equation for two-dimensional weakly modulated gravity waves on deep water. A key ingredient in this derivation is a Birkhoff normal form transformation that eliminates all non-resonant cubic terms and allows for a non-perturbative reconstruction of the free surface. This modulational approximation is tested against numerical solutions of the classical Dysthe's equation and against direct numerical simulations of Euler's equations for nonlinear water waves. An alternate spatial form is also proposed and tested against laboratory experiments on short-wave packets. (joint work with W. Craig, P. Guyenne, A. Kairzhan, B. Xu)
March 15th, 2021
Peter Miller (Department of Mathematics, University of Michigan)
Title: Universal Wave Breaking in the Semiclassical Sine-Gordon Equation
Abstract: The sine-Gordon equation has slowly-modulated librational wave solutions that are approximated at leading-order by a Whitham averaging formalism. The Whitham modulation equations are an elliptic quasilinear system whose solutions develop singularities in finite time. We show that when the solution of the Whitham system develops a generic type of gradient catastrophe singularity, the solution of the sine-Gordon equation locally takes on a universal form, independent of initial data and described in terms of the real tritronquée solution of the Painlevé-I equation and a two-parameter family of exact solutions of sine-Gordon that represent space-time localized defects on an otherwise periodic background wave. This is joint work with Bing-Ying Lu.
March 29th, 2021
Gino Biondini (Department of Mathematics, State University of New York at Buffalo)
Title: Time evolution problems for (2+1)-dimensional evolution equations: Whitham modulation theory and its applications
Abstract: In 1965, Whitham formulated the eponymous nonlinear modulation theory for the Korteweg-deVries (KdV) equation. Whitham's theory allows one to derive a set of hyperbolic PDEs describing modulations of the traveling-wave solutions of the KdV equation. The theory was later generalized to many other systems, and has been applied with great success in large a variety of settings. Most studies, however, have been limited to (1+1)-dimensional systems, i.e., PDEs in one spatial dimension. After a brief review of the Kadomtsev-Petviashvili (KP) equation and its solutions, I will show how one can formulate Whitham modulation theory for (2+1)-dimensional equations and derive the Whitham modulation equations for the KP equation as well as other equations of KP type. I will then discuss some key properties of the KP-Whitham system of equations. Finally, I will show how the system can be successfully used to characterize analytically for the first time the evolution of a variety of initial conditions, including partial soliton stems and a combination of solitons and a mean flow.
April 12th, 2021
Boris Malomed (Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University)
Title: Nonlinear dynamics of wave packets in tunnel-coupled harmonic-oscillator traps
Abstract: We consider a two-component linearly-coupled system with the cubic nonlinearity and harmonic-oscillator (HO) confining potential. In the symmetric system, with the HO trap acting in both components, we consider Josephson oscillations (JO) initiated by an input in the form of the HO's ground state (GS) or dipole mode (DM), launched in one component. With the increase of the strength of the self-focusing nonlinearity, spontaneous symmetry breaking (SSB) between the components takes place in the JO state. Under still stronger nonlinearity, the regular JO initiated by the GS input carry over into a chaotic dynamical state. For the DM input, the chaotization happens at smaller powers, followed by SSB. In the system with the defocusing nonlinearity, the chaotization occurs in a small area of the parameter space. In the half-trapped system, with the HO potential applied to a single component, we address the spectrum of confined modes of the linearized system. The spectrum is found analytically for weak and strong inter-component coupling, and numerically in the general case. Under the action of the coupling, the existence region of the confined modes shrinks for GSs and expands for DMs. In the nonlinear system, the existence region for confined modes is found numerically and by means of the Thomas-Fermi approximation. Lastly, particular exact analytical solutions for confined modes, including vortices, are found in 1D and 2D asymmetric linearized systems. They represent bound states in the continuum.
April 26th, 2021
Christopher K.R.T. Jones (Renaissance Computing Institute, University of North Carolina at Chapel Hill)
Title: Counting Gap Eigenstates for the 3D Radial Gross-Pitaevskii Equation by a Maslov Index
Abstract: In basic quantum mechanics, Sturm-Liouville theory gives us a convenient way to count eigenvalues on the real axis of a linear Schrödinger equation on the line with decaying potential. The count can be expressed in terms of an angle varying in the phase space of the eigenvalue problem. Adding such a potential to a cubic NLS, as in the GP equation, leads to a more complex problem (no pun intended.) For various reasons, it is important to count the eigenstates in the gap on the imaginary axis below the essential spectrum. The question is whether there is an analogue of the angular variation in the appropriate phase space. We show how this can be realized (again, no pun intended) through the Maslov Index. There is a seemingly magical formula of Arnold’s that will be explained as it is helpful on giving a perspective to the calculations.
Joint work with Dmitro Golovanich and Jeremy Marzuola (both in Math, UNC-CH)
May 10th, 2021
Arnd Scheel (School of Mathematics, University of Minnesota)
Title: Propagation into unstable states: linear predictions and nonlinear results
Abstract: I will describe recent progress in the analysis of front propagation into unstable states from a perspective of coherent structures. In many cases, linear criteria predict front speed and even the selected state in the wake of the front well. The talk will outline these linear predictions and a conceptual framework that yields nonlinear results corresponding to those linear predictions thus confirming the "marginal stability conjecture". The main idea is to phrase the invasion problem as a tail cut-off problem for a critical invasion front, which leads to supercritical perturbations in a diffusive stability problem. We capture the leading order effect of these perturbations in a log-shifted coordinate frame using sharp resolvent estimates for diffusive decay in the leading edge. I'll also point to limitations of the results and their relation to front invasion driven by more general complex resonances in the dispersion relation. The talk is based on joint work with Montie Avery, Gregory Faye, and Matt Holzer.
June 7th, 2021
Jared Bronski (Department of Mathematics, University of Illinois at Urbana-Champaign)
Title: Novel Instabilities in regularized long-wave equations
Abstract: In the usual long-wave approximations for water waves, such as the Korteweg-de Vries and related equations, instability is a finite to long wavelength phenomenon, in that outside of a bounded set in the eigenvalue plane the spectrum of the linearization about a traveling wave solution lies on the imaginary axis. However for a number of “regularized” long-wavelength equations including the regularized Boussinesq, Benney-Luke, and Bona-Chen-Saut system the relatively weak dispersion leads to an interesting new phenomenon. We find that the behavior of the spectrum depends sensitively on the parameters of the traveling wave: for some sets of parameters the eigenvalues lie along the imaginary axis, while for other, nearby values of the parameters the eigenvalues tend to infinity along some curve with non-vanishing real part. We have a high-frequency (short-wavelength) asymptotic theory to explain this phenomenon, which gives excellent agreement with numerical simulations. Joint work with Vera Mikyoung Hur and Samuel Wester (Illinois).
June 21st, 2021
Gigliola Staffilani (Department of Mathematics, Massachusetts Institute of Technology)
Title: Energy transfer for solutions to the nonlinear Schrodinger equation on irrational tori
Abstract: In this talk I will outline some results on the study of transfer of energy for solutions to the periodic 2D (torus domain) cubic defocusing nonlinear Schrodinger equation. In particular I will focus on the differences of the dynamics of solutions in the rational versus irrational torus. Some numerical experiments will also be presented. (The most recent work presented is in collaboration with A. Hrabski, Y. Pan and B. Wilson.)
July 5th, 2021
Alan R Champneys (University of Bristol)
Title: Localised patterns and semi-strong interaction, a unifying framework for reaction-diffusion systems
Abstract: Systems of activator-inhibitor reaction-diffusion equations posed on an infinite line are studied using a variety of analytical and numerical methods. A canonical form is considered that contains all known models with simple cubic autocatalytic nonlinearity and arbitrary constant and linear kinetics. Restricting attention at first to models that have a unique homogeneous equilibrium, this class includes the classical Schnakenberg and Brusselator models, as well as other systems proposed in the literature to model morphogenesis. Such models are known to feature Turing instability, when activator diffuses more slowly than inhibitor, leading to stable spatially periodic patterns. Conversely in the limit of small feed rates, semi-strong interaction asymptotic analysis as introduced by Michael Ward and his collaborators shows existence of isolated spike-like patterns.
Connecting these two regions, a certain universal two-parameter state diagram is revealed in which the Turing bifurcation becomes sub-critical, leading to the onset of homoclinic snaking. This regime then morphs into the spike regime, with the outer-fold being predicted by the semi-strong asymptotics. A rescaling of parameters and field concentrations shows how this state diagram can be studied independently of the diffusion rates. Temporal dynamics is found to strongly depend on the diffusion ratio though. A Hopf bifurcation occurs along the branch of stable spikes, which is subcritical for small diffusion ratio, leading to collapse to the homogeneous state. As the diffusion ratio increases, this bifurcation typically becomes supercritical, interacts with the homoclinic snaking and also with a supercritical homogeneous Hopf bifurcation, leading to complex spatio-temporal dynamics. The details are worked out for a number of different models that fit the theory using a mixture of weakly nonlinear analysis, semi-strong asymptotics and different numerical continuation algorithms.
The theory is extended include models, such as Gray-Scott, with bistability of homogeneous equilibria. A homotopy is studied that takes a Schnakenberg-like glycolysis model for r = 0 to the Gray-Scott model for r = 1. Numerical continuation is used to understand the complete sequence of transitions to two-parameter bifurcation diagrams within the localised pattern parameter regime as r varies. Several distinct codimension-two bifurcations are dis-covered including cusp and quadruple zero points for homogeneous steady states, a degenerate heteroclinic connection and a change in connectedness of the homoclinic snaking structure. The analysis is repeated for the Gierer-Meinhardt system, which lies outside the canonical framework. Similar transitions are found under homotopy between bifurcation diagrams for the case where there is a constant feed in the active field, to it being in the inactive field. Wider implications of the results are discussed for other kinds of pattern-formation systems as well as to distinguishing between different kinds of observed behaviour in the natural world.
September 14th, 2021
Keith Promislow (Department of Mathematics, Michigan State University)
Title: Entropy, Packing, and Patterns
Abstract: This talk is a series of vignettes about reality poking holes in my intuition. Some are very simple, all involve efforts to incorporate entropy and constraints associated to physical proximity into models of pattern formation. The underlying theme is that entropy and packing impose constraints on viable models while imbuing them with rich structure, often pushing them in directions that are more singular and less controllable than we may wish. Primary applications are to a simple model of the packing of soft balls, to brine channel formation in sea ice, and the role of charge in polymer membranes.
September 28th, 2021
Michael Ward (Department of Mathematics, University of British Columbia)
Title: Topics in Localized Pattern Formation for Reaction-Diffusion Systems in 3-D
Abstract: Localized spot patterns, where one or more solution components concentrates at certain points in the domain, are a common class of localized pattern for reaction-diffusion systems, and they arise in a wide range of modeling scenarios. Although there is a rather well-developed theoretical understanding for this class of localized pattern in one and two space dimensions, a theoretical study of such patterns in a 3-D setting is, largely, a new frontier. In an arbitrary bounded 3-D domain, the existence, linear stability, and slow dynamics of localized multi-spot patterns is analyzed for the well-known singularly perturbed Gierer-Meinhardt (GM) activator-inhibitor and Schnakenberg systems in the limit of a small activator diffusivity. Depending on the range of parameters, spot patterns can undergo competition instabilities, leading to spot-annihilation events, or shape-deforming instabilities triggering spot self-replication events. In the absence of these instabilities, the spot locations evolve slowly towards their equilibrium locations according to an ODE gradient flow, which is determined by a discrete energy involving the reduced-wave Green's function. The central role of a certain core problem, which characterizes the profile of a localized spot, on the solution behavior is emphasized. Open problems for localization on higher co-dimension structures, such as stripes and filaments, are discussed.
October 12th, 2021
Jens Rademacher (Department of Mathematics, University of Bremen)
Title: Dynamics of some strongly interacting localised non-linear waves
Abstract: Numerical simulations show intriguing space-time dynamics of strongly interacting pulses, for instance in FitzHugh-Nagumo-type equations. Here strong interaction refers to pulse collisions, replications etc. in contrast to semi-strong or weak interaction in which some large distance is required. A rigorous analysis of these phenomena appears out of reach for systems of such equations, which are required for stability of pulses therein.
In this talk I discuss some insights into strong interactions for simplified settings. On the one hand, the theta-equations from neural dynamics are scalar equations with circular local phase space, for which the comparison principle admits to track strong interaction of kinks as an analogue of pulses. On the other hand, the Greenberg-Hastings-cellular automata are a model for excitable media that even admit a study of ergodic properties, such as the topological entropy, of annihilating pulses.
This is joint work with Marc Kesseböhmer, Antoine Pauthier and Dennis Ulbrich.
October 26th, 2021
Joceline Lega (Department of Mathematics, University of Arizona)
Title: A dynamical systems view of special solutions to the discrete Painlevé 1 equation
Abstract: The discrete Painlevé 1 equation (dP1) is a one-dimensional, second-order, non-autonomous, discrete dynamical system. Its non-polar solutions, which grow monotonically and without bounds as the number of iterations (n) increases, have relevance for orthogonal polynomials and certain combinatorial problems. In this talk, I will introduce this system and describe our numerical investigation of its solutions. In particular, I will present a change of variables that transforms specific non-polar solutions of dP1 into heteroclinic orbits between two finite fixed points. I will characterize the dynamics in the vicinity of these fixed points and explain how these results may be used to obtain an asymptotic expansion describing how the solutions in the original coordinates behave as n goes to infinity.
This is joint work with Nick Ercolani and Brandon Tippings. A preprint is available online at https://arxiv.org/abs/2109.03384.
November 23rd, 2021 (at 9:00am, Eastern Time)
Sergej Flach (Center for Theoretical Physics of Complex Systems, Institute for Basic Science Daejeon)
Title: Lyapunov spectrum scaling for many-body dynamics close to integrability
Abstract: We use unitary map evolution to address the thermalization of many-body dynamical systems close to an integrable limit. Integrable limits are classified as either long-range or short-range, depending on the connectivity between actions. The connectivity type is imposed by the nonintegrable perturbation. We compute the Lyapunov spectra and analyze their scaling properties. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the largest Lyapunov exponent being the only control parameter. Short-range networks have an additional second scaling parameter which describes the exponential suppression of all relative to the largest Lyapunov exponent.
This is a joint work with Merab Malishava, and a preprint is available at arXiv:2109.01361
December 7th, 2021
Vera Mikyoung Hur (Department of Mathematics, University of Illinois at Urbana-Champaign)
Title: Stable undular bores
Abstract: I will discuss the asymptotic stability of the traveling front solutions to the Korteweg-de Vries--Burgers equation, and for a general class of dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the front, for relatively weak dispersion effects. We instead exploit the temporal modulation of the translation parameter of the front solution, establishing our stability criterion that a Schrodinger operator in one dimension has exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of a Bargman-type functional, related to the area between the front and the corresponding ideal shock. We analytically verify that our stability criterion is met for an open set including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski and Zhao Yang.
December 21st, 2021
Annalisa Calini (Department of Mathematics, College of Charleston)
Title: Homoclinic Orbits as Rogue Waves in Nonlinear Schrodinger Models
Abstract: I will describe not-so-recent work with Constance Schober on homoclinic orbits for uniform and quasi-periodic background states in nonlinear Schrodinger models of waves in deep water. The main thrusts are examining linear stability/instability, the effects of phase coalescence on wave amplification, and persistence under physically relevant perturbations.
January 11th, 2022
Jianke Yang (The University of Vermont)
Title: Pattern Formation in KP-I Lumps
Abstract: Pattern formation in higher-order lumps of the Kadomtsev-Petviashvili I equation at large time is analytically studied. For a broad class of these higher-order lumps, we show that two types of solution patterns appear at large time. The first type of patterns comprise fundamental lumps arranged in triangular shapes, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. As time evolves from large negative to large positive, this triangular pattern reverses itself along the x-direction. The second type of patterns comprise fundamental lumps arranged in non-triangular shapes in the outer region, which are described analytically by nonzero-root structures of the Wronskian-Hermit polynomials, together with possible fundamental lumps arranged in triangular shapes in the inner region, which are described analytically by root structures of the Yablonskii-Vorob'ev polynomials. When time evolves from large negative to large positive, the non-triangular pattern in the outer region switches its x and y directions, while the triangular pattern in the inner region, if it arises, reverses its direction along the x-axis. Our predicted patterns at large time are compared to true solutions, and excellent agreement is observed. This is joint work with Dr. Bo Yang.
January 25th, 2022
Alexander Mielke (Weierstraß-Institut für Angewandte Analysis und Stochastik and Humboldt-Universität zu Berlin)
Title: On the longtime behavior of solutions to a coupled degenerate parabolic system motivated by thermodynamics
Abstract: We discuss a system of two coupled parabolic equations that have degenerate diffusion constants depending on the energy-like variable. The dissipation of the velocity-like variable is fed as a source term into the energy equation leading to conservation of the total energy. The motivation of studying this system comes from Prandtl's and Kolmogorov's one and two-equation models for turbulence, where the energy-like variable is the mean turbulent kinetic energy. After providing a thermodynamically motivated gradient structure we establish convergence into steady state for bounded domains and provide a conjecture on the self-similar longtime behavior of the solutions in $ R^d $.
February 8th, 2022
Arjen Doelman (Leiden University)
Title: Spatial Ecology & Singularly Perturbed Reaction-Diffusion Equation
Abstract: Pattern formation in ecological systems is driven by counteracting feedback mechanisms on widely different spatial scales. Moreover, ecosystem models typically have the nature of reaction-diffusion systems: the dynamics of ecological patterns can be studied by the methods (geometric) singular perturbation theory. In this talk we give an overview of the surprisingly rich cross-fertilization between ecology, the physics of pattern formation and the mathematics of singular perturbations. We show how mathematical insights uncover mechanisms by which ecosystems may evade (catastrophic) tipping under slowly varying circumstances – based on the validation of these insights by field observations. Vice versa, ecosystem models motivate the study of classes of singularly perturbed reaction-diffusion equations that exhibit much more complex behavior than ‘classical’ Gray-Scott/Gierer-Meinhardt type models. By considering the so-called ‘slow patterns’, we take the first step in the analysis of these models.
February 22th, 2022
Wolf-Juergen Beyn (Bielefeld University)
Title: Stability and computation of traveling oscillating fronts in complex Ginzburg-Landau equations
Abstract: Traveling oscillating fronts (TOFs) are specific one-dimensional waves which occur in complex Ginzburg-Landau equations, for example of quintic type. A TOF has a profile which connects the zero state to a nonzero rest state; it travels in space while simultaneously rotating its complex phase. In this talk we present two results on the nonlinear stability of TOFs with asymptotic phase. The first one ensures exponential decay in time for an initial perturbation which is the sum of an exponentially localized part and a front-like part approaching a small but nonzero limit at infinity. The second one ensures algebraic decay in time and allows initial perturbations to be small in a polynomially weighted norm. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion. Our results are based on two different approaches to derive nonlinear stability from linearized stability in such a sensitive setting. If time permits, we will show convergence of the freezing method which allows, during simulation near a TOF, to adaptively separate the emerging profile in a fixed frame from the dynamics on the underlying two-dimensional symmetry group.
This talk is based on a joint work with Christian Doeding (Ruhr-Unversity Bochum).
March 22th, 2022
Philip Rosenau (School of Mathematics, Tel Aviv University)
Title: On multi-dimensional compact solitary patterns
Abstract: As though to compensate for the rarity of multidimensional integrable systems in higher dimensions, spatial extensions of many of the well-known nonlinear dispersive equations on the line, exhibit a remarkably rich variety of solitary patterns unavailable in 1-D. Our work systematizes this observation with a special attention paid to compactons - solitary waves with compact support - where this effect is found to be far more pronounced and begets a zoo of multi-dimensional compact solitary patterns.
One manifestation of this phenomenon is found in the sublinear NLS and Complex Klein-Gordon where the compactons inducing mechanism coupled with azimuthal spinning may expel the compact vortices from the origin to form a finite or countable number of genuine ring-vortices. Such rings are an exclusive feature of compacton supporting systems.
April 5th, 2022
Hermen Jan Hupkes (University of Leiden)
Title: Waves in Inhomogeneous Media
Abstract: We discuss techniques to establish the (meta)-stability of wavefronts travelling through inhomogeneous media. The inhomogeneities can be caused by the presence of noise, but can also occur naturally as a consequence of the spatial discreteness of the underlying system. By carefully tracking the temporal-spatial evolution of the phase of the wavefront, we can control large perturbations from deterministic (planar) waves. In particular, our phase evolutions are described by stochastic differential equations or discrete curvature flows.
This is a joint work with Mia Jukic and Christian Hamster.
April 19th, 2022
Matthew Johnson (University of Kansas)
Title: Stability of periodic Lugiato-Lefever Waves
Abstract: In this talk I will describe recent advances in the stability analysis of T-periodic stationary solutions of the Lugiato-Lefever equation, a damped nonlinear Schrödinger type equation with forcing that arises in nonlinear optics. Several recent works have studied the stability of such waves to so-called ‘subharmonic’ perturbations, i.e. NT-periodic perturbations for some natural number N. However, these results are degenerate for large N since both the rate of decay of perturbations, along with the domain of attraction, both tend to zero as N tends to infinity. In this talk, I will discuss a new methodology for obtaining subharmonic stability results for the LLE which are uniform in N. This approach is quite general, and applies to a wide class of dissipative models.
May 17th, 2022
Constance Schober (University of Central Florida)
Title: Nonlinear damped spatially periodic breathers and the emergence of soliton-like rogue waves
Abstract: The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger equation, prominent in modeling rogue waves, are unstable. In this talk we numerically examine the routes to stability of the SPBs and related rogue wave activity in the framework of a nonlinear damped higher order nonlinear Schrödinger (NLD-HONLS) equation. The NLD-HONLS solutions are initialized with single-mode and two-mode SPB data at different stages of their development. The Floquet spectral theory of the NLS equation is applied to interpret and provide a characterization of the perturbed dynamics in terms of nearby solutions of the NLS equation. A broad categorization of the routes to stability of the SPBs is determined. Novel behavior related to the effects of nonlinear damping is obtained: tiny bands of complex spectrum develop in the Floquet decomposition of the NLD-HONLS data, indicating the breakup of the SPB into either a one or two ``soliton-like'' structure. For solutions initialized in the early to middle stage of the development of the modulational instability, we find that all the rogue waves in the NLD-HONLS flow occur when the spectrum is in a one or two soliton-like configuration. When the solutions are initialized as the modulational instability is saturating, rogue waves may occur when the spectrum is not in a soliton-like state. Another distinctive feature of the nonlinear damped dynamics is that the growth of instabilities can be delayed and expressed at higher order due to permanent frequency downshifting.
May 31st, 2022
Natalia Berloff (University of Cambridge)
Title: Unconventional computing with liquid light
Abstract: The recent advances in developing physical platforms for solving combinatorial optimisation problems reveal the future of high-performance computing for quantum and classical devices. Unconventional computing architectures were proposed for numerous optical systems, memristors, lasers and nanolasers, optoelectronic systems, polariton and photon condensates. A promising approach to achieving computational supremacy over the classical von Neumann architecture explores classical and quantum hardware as Ising and XY machines. Gain-dissipative platforms such as the networks of optical parametric oscillators, coupled lasers and non-equilibrium Bose-Einstein condensates such as exciton-polariton or photon condensates use an approach to finding the global minimum of spin Hamiltonians which is different from quantum annealers or quantum computers. In my talk, I will discuss the principles of the operation of the devices based on such systems, and the challenges they present, with a particular focus on polariton graph simulators.
June 14th, 2022
Nathan Kutz (University of Washington)
Title: The Future of Governing Equations
Abstract: Machine learning and AI algorithms are transforming a diverse number of fields in science and engineering. This is largely due their success in model discovery which turns data into reduced order models and neural network representations that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. We introduce a number of data-driven strategies, including targeted uses of deep learning, for discovering nonlinear multiscale dynamical systems, compact representations, and their embeddings from data. Importantly, data-driven architectures must jointly discover coordinates and parsimonious models in order to produce maximally generalizable and interpretable models of physics-based systems and processes.
June 28th, 2022
Guido Schneider (Universität Stuttgart)
Title: Failure of amplitude equations
Abstract: Approximation by modulation, envelope or amplitude equations plays a fundamental role in the understanding of complex systems. Famous examples are the KdV approximation of the water wave problem, the Ginzburg-Landau approximation for pattern-forming systems, or the NLS approximation in nonlinear optics. In spite of faster and faster computers they are still the starting point of the analysis of the qualitative behavior of many complex systems. It is widely believed that the formally derived amplitude equations always make correct predictions about the dynamics of the original system. In fact, recent decades have seen proofs of many rigorous approximation results. However, there are many pairs of amplitude equations and classes of original systems for which it has not yet been possible to establish such an approximation result. In this talk we would like to concentrate on results that show that in various situations formally derived amplitude equations do NOT make correct predictions about the dynamics of the original system.
September 14, 2022
Bob Pego (Department of Mathematical Sciences, Carnegie Mellon University)
Title: Nonlinear waves and photon loss in the Kompaneets model of Compton scattering
Abstract: The Kompaneets equation is fundamental for explaining the Sunyaev-Zeldovich effect, which involves deformation of the cosmic microwave background energy spectrum due to Compton scattering of photons by electrons. We establish L1 convergence to Bose-Einstein equilibria in large time and prove several results on the existence and behavior of a `condensate' of photons at the zero-energy boundary. The Kompaneets equation is a scalar conservation law with a degenerate parabolic nature that permits a loss of photons in finite time due to shock formation at zero energy. Solutions satisfy entropy decay, an Oleinik one-sided slope bound, and L1 contraction and comparison principles.
October 12, 2022
Mark Hoefer (Department of Applied Mathematics, University of Colorado, Boulder)
Title: Breathers as the nonlinear superposition of solitary and cnoidal-like waves
Abstract: Integrable systems such as the Korteweg-de Vries equation exhibit two classes of degenerate 2-phase breather solutions that can be viewed as nonlinear superpositions between a soliton and a periodic (e.g., cnoidal) traveling wave. These solutions are explicitly obtained in terms of Jacobi theta functions using a Darboux transformation. KdV breather properties are described in terms of the underlying Lax spectrum. Using a simple fluid experiment and fixed point computations of a space-time boundary value problem, the persistence of this nonlinear superposition principle to a physical, non-integrable setting is demonstrated. Cnoidal-like, locally periodic traveling waves are experimentally generated via a wave maker and shown to overtake or be overtaken by a soliton on a constant background by forming a breather state consisting of a localized, moving topological defect within the cnoidal-like wave. Both dark and bright breathers as well as their interactions are measured. The existence of breather solutions to the corresponding nonlinear, long-wave model of the fluid experiment is demonstrated by numerical solution of a space-time boundary value problem.
November 9, 2022
Kevin Zumbrun (Department of Mathematics, Indiana University Bloomington)
Title: Large-amplitude modulation of periodic traveling waves
Abstract: We introduce a new approach to the study of modulation of high-frequency periodic wave patterns, based on pseudo-differential analysis, multi-scale expansion, and Kreiss symmetrizer estimates like those in hyperbolic and hyperbolic-parabolic boundary-value theory. Key ingredients are local Floquet transformation as a preconditioner removing large derivatives in the normal direction of background rapidly oscillating fronts and the use of the periodic Evans function of Gardner to connect spectral information on component periodic waves to block structure of the resulting approximately constant-coefficient resolvent ODEs. Our main result is bounded-time existence and validity to all orders of large-amplitude smooth modulations of planar periodic solutions of multi-D reaction diffusion systems in the high-frequency/small wavelength limit.
This result is new for either large amplitudes or multi-d. Applications to pattern formation are many, and well known (Howard-Kopell, Doelman-Sandstede-Scheel-Schneider, ...). The method suggests possibilities also to treat more complicated phenomena in shallow-water and thin-film flow. The talk is based on a joint work with G. Metivier.
December 14, 2022
Mariana Haragus (Département de Mathématiques, UFR des Sciences et Techniques, Université de Franche-Comté)
Title: Transverse dynamics of line periodic water waves
Abstract: Line periodic water waves are solutions of the three-dimensional water wave problem which are periodic in one horizontal coordinate and do not depend on a second, transverse, horizontal coordinate. For such waves, the transverse stability question is concerned with their stability with respect to three-dimensional perturbations, hence also depending on the horizontal coordinate in which the line periodic waves are constant. Relying upon an abstract, rather simple, linear instability criterion, we show that capillary-gravity periodic water waves are transversely unstable in several parameter regimes. As a consequence, a dimension-breaking bifurcation occurs in which doubly periodic waves bifurcate from a transversely unstable line periodic wave.
January 11, 2023
Peter Constantin (Princeton University)
Title: On the Nernst-Planck-Navier-Stokes System
Abstract: The Nernst-Planck-Navier-Stokes system describes the evolution of ionic species interacting with fluids. I will present some background, recent results and open questions in the area.
February 8, 2023
David Ambrose (Drexel University)
Title: Dependence on the domain in parabolic dynamics
Abstract: We will discuss some nonlinear systems of parabolic equations, all of which exhibit significant differences in dynamics depending on the domain. To begin we will give some background on the Kuramoto-Sivashinsky equation, which can exhibit decay, stable coherent structures, or chaotic dynamics depending on the size of periodic cell taken as the domain. We will then give two examples of parabolic equations which have different behavior for decaying solutions and for periodic solutions: the Constantin-Lax-Majda equation with diffusion, and the Navier-Stokes equations. For the Constantin-Lax-Majda equation with diffusion, the difference is with respect to the question of global existence versus singularity formation. For the Navier-Stokes equations, we demonstrate an improved gain of regularity for solutions on the torus. This includes joint work with Pavel Lushnikov, Anna Mazzucato, Milton Lopes Filho, Helena Nussenzveig Lopes, Michael Siegel, and Denis Silantyev.
March 8, 2023
Peter Bates (Michigan State University)
Title: Existence and Persistence of Invariant Manifolds for PDE's and SPDE's
Abstract: We see dynamic coherent structures everywhere we look in our universe, as well as in experiments, and in simulations. Sometimes there is an intermittency or seemingly periodic distortions in the patterns. Sometimes patterns persist for a long time, then transition to less complex patterns until it seems, stabilization occurs. We may suspect that these patterned states lie on an invariant manifold of states, where the dynamics can be more easily understood, and perhaps can be reliably predicted.
One can easily be fooled by simulations, however, and one would like to have some assurance that the conjectured structure actually exists. This is made more challenging when the system is infinite-dimensional, as with PDE's of evolution, and even more so when stochastic perturbations are included. In the latter case, the question of `what would an invariant manifold look like, i.e., how should it be described?' must be answered.
In this talk, I will describe conditions under which a global invariant manifold of spike states exists for $$u_t = \e^2\Delta u -u +f(u) +\eta g(\theta_t\omega,u)$$ with homogeneous Neumann boundary conditions in a smoothly bounded domain $\Omega$. Here, $f(0)=f'(0)=0, \; g(\omega,u)$ is $C^1$ in $u$ for fixed $\omega$ and measurable in $\omega$ for fixed $u$. $\theta_t$ is a metric dynamical system on a probability space, so $\theta_t\omega$ models the randomness of perturbations.
This is based on work with Kening Lu, Chongchun Zeng, Linfeng Zhou, Weinian Zhang, and Ji Li.
April 12, 2023
Nancy Rodriguez (University of Colorado Boulder)
Title: On local and non-local diffusion-advection-reaction models for animal movement
Abstract: A successful wildlife management plan relies on two key factors: (1) the understanding of driving factors influencing the movement of social animals and (2) the understanding of what movement strategies are optimal depending on the environment. In this talk, I will first discuss results from work focused on determining how some social animals, such as Meerkats, move. We present a non-local reaction-advection-diffusion model along with an efficient numerical scheme that enables the incorporation of data. The second part of the talk will focus on how directed movement can help species overcome the strong Allee effect on both bounded and unbounded domains. I will also discuss the connection to optimal movement strategies in the context of the strong Allee effect.
May 10, 2023
Paris Perdikaris (University of Pennsylvania)
Title: Resolution-invariant Generative Models for Functional Data
Abstract: Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation. We also show that conditional formulations of VANO can be used to tackle inverse problems where not all functional outputs of a system are available for observation.
Joint work with: Jacob H. Seidman, Georgios Kissas and George J. Pappas
June 14, 2023
Bernard Deconinck (University of Washington)
Title: The instabilities of Stokes Waves
Abstract: Stokes Waves are traveling wave solutions of the Euler Water Wave problem. In this talk, I will provide an overview of the different instabilities that govern the dynamics of their perturbations. For small-amplitude waves (near flat water) an abundance of studies this past decade have resulted in a fairly complete understanding. For high-amplitude waves (near the limiting 120 degree wave) so far only numerical results are available. I will finish the talk with a plethora of conjectures on the nature of the instability spectra of near-extreme Stokes waves.
October 2nd, 2023
Thomas Bridges (Department of Mathematics, University of Surrey)
Title: Travelling wave solutions connecting distinct periodic orbits in Hamiltonian PDEs
Abstract: Travelling wave solutions of Hamiltonian PDEs on the real line, asymptotic to distinct periodic solutions at plus and minus infinity are considered. This problem blends the theory of travelling fronts with the theory of symplectic periodic orbits. From the former, jump conditions are derived, and from the latter Hamiltonian bifurcation theory is applied to the states at infinity. The normal form for a codimension two singularity of the periodic orbits is used to organise the bifurcations at infinity. Other topics of interest include (a) the role of Whitham theory in deducing jump conditions, (b) replacing the periodic orbits at infinity by invariant tori, (c) the role of the Maslov index of the periodic orbits at infinity, and (d) disjoint speeds of the states at infinity. Numerical results are presented for traveling waves of model equations from the theory of water waves. The results include the construction of hydraulic jumps with oscillatory tails. This talk is based on joint work with David Lloyd, Dan Ratliff, Michael Shearer, and Pat Sprenger.
Reference: P. Sprenger, T.J. Bridges, M. Shearer [2023] Traveling wave solutions of the Kawahara equation joining distinct periodic waves, J. Nonlinear Sci 33: 79.
November 6th, 2023
Gennady El (Mathematics, Physics and Electrical Engineering, Northumbria University)
Title: Emergent hydrodynamics of soliton gases in nonlinear dispersive waves
Abstract: Soliton gases represent large random ensembles of interacting solitons that display non-trivial emergent, macroscopic, behaviours ultimately determined by the properties of the elementary two-soliton collisions. Originally introduced by V. Zakharov in 1971 the concept of soliton gas has recently attracted significant interest from both mathematics and physics communities. The large-scale evolution of non-equilibrium dense soliton gases in integrable dispersive systems is described by a nonlinear integro-differential kinetic equation for the density of states in the spectral (Lax) phase space. In my talk, I will outline the main ideas of the spectral theory of soliton gases and its applications to a range of fundamental dispersive hydrodynamic phenomena: from modulational instability to dispersive shock waves.
December 4th, 2023
Laurette Tuckerman (Sorbonne Université)
Title: Exotic patterns of Faraday waves
Abstract: A standing wave pattern appears on the free surface of a fluid layer when it is subjected to vertical oscillation of sufficiently high amplitude. Like Taylor-Couette flow (TC) and Rayleigh-Benard convection (RB), the Faraday instability is one of the archetypical pattern-forming systems. Unlike TC and RB, the wavelength is controlled by the forcing frequency rather than by the fluid depth, making it easy to destabilize multiple wavelengths everywhere simultaneously. Starting in the 1990s, experimental realizations using this technique produced fascinating phenomena such as quasipatterns and superlattices which in turn led to new mathematical theories of pattern formation. Another difference is that the Faraday instability has been the subject of surprisingly little numerical study, lagging behind TC and RB by several decades. The first 3D simulation reproduced hexagonal standing waves, which were succeeded by long-time recurrent alternation between quasi-hexagonal and beaded striped patterns, interconnected by spatio-temporal symmetries. In a large domain, a supersquare is observed in which diagonal subsquares are synchronized. A liquid drop subjected to an oscillatory radial force comprises a spherical version of the Faraday instability. Simulations show Platonic solids alternating with their duals while drifting.
January 8th, 2024
Paul Milewski (The Pennsylvania State University)
Title: Resonant free-surface water waves in cylinders
Abstract: Nonlinear resonance is a mechanism by which energy is continuously exchanged between a small number of linear wave modes, and is common to many nonlinear dispersive wave systems. In the context of free-surface gravity waves, nonlinear resonances have been studied extensively over the past 60-years, almost always on domains that are large compared to the characteristic wavelength (such as oceans). In this case, the dispersion relation dictates that only quartic (4-wave) resonances may occur. In contrast, nonlinear resonances in confined three-dimensional geometries have received relatively little attention, where, perhaps surprisingly, 3-wave resonances do occur. We will present the results characterizing the configuration and dynamics of resonant triads in cylindrical basins of arbitrary cross sections, demonstrating that these triads are ubiquitous.
February 5th, 2024
Triantaphyllos Akylas (MIT)
Title: Nonlinear Surface Wave Patterns and Exponential Asymptotics
Abstract: The radiation of one-dimensional (1D) steady surface gravity waves by a uniform stream U over locally confined (width L) smooth topography is analyzed asymptotically in the low-Froude-number limit, F = U/sqrt(gL) << 1. In this regime, the radiated wave amplitude, although formally exponentially small with respect to F, is determined by a fully nonlinear mechanism even for small topography amplitude. This mechanism controls the wave response for a broad range of flow conditions, in contrast to linear theory which has very limited validity. Furthermore, the present asymptotic methodology can be extended to 2D wave patterns in 3D flow, including the Kelvin ship-wave problem. (This is joint work with Takeshi Kataoka, Kobe University, Japan.)
March 4th, 2024
Svetlana Roudenko (Florida International University)
Title: Solitary waves in the KdV-type equations: stability vs. instability & blow-up
Abstract: In this talk we start with a higher-dimensional generalization of the well-known shallow water wave model of Korteweg-de Vries (KdV), called Zakharov-Kuznetsov (ZK) equation, which originally was proposed to model the weakly nonlinear ion-acoustic waves in a uniform magnetic field. We review the dynamics of ZK solitary waves, in particular, the asymptotic stability in the 3D standard model and blow-up in the 2D critical model. We then consider generalizations to nonlocal dispersion, the fractional gKdV/ZK equations such as HBO or Shrira models, which, for instance, in the 2D setting describe long-wave perturbations in a boundary layer type shear flow in the ocean, and examine stability and instability of solitary waves there. Finally, we focus on a specific example of the 1D generalized Benjamin-Ono equation and discuss stability of solitary waves and blow-up.
(This talk is based on joint works with L.G. Farah, J. Holmer, K. Yang, O. Riaño, C. Klein, N. Stoilov.)
April 15th, 2024
Walter Strauss (Division of Applied Mathematics, Brown University)
Title: Transverse Instability of Stokes Waves
Abstract: It was discovered in the 1960s, numerically and heuristically, that even very small Stokes water waves are unstable when subject to long-wave longitudinal perturbations. This is the modulational (Benjamin-Feir) instability. Around 1980 it was discovered numerically that they are also unstable when subject to transverse (3D) co-periodic perturbations. I will present the first proof of this phenomenon. It is joint work with Huy Nguyen and Ryan Creedon.
What we prove is the spectral instability. The fluid is allowed to have either infinite or finite depth. A conformal mapping is used to fix the free boundary. The linearized operator is pseudo-differential. It is analytically expanded in the wave amplitude (epsilon) and the perturbation parameter (delta). Very lengthy calculations are required to find the unstable eigenvalue, which occurs at third order in epsilon. I will emphasize the methods and spare the audience the long calculations.
May 6th, 2024
Todd Kapitula (Department of Mathematics and Statistics, Calvin University)
Title: Mentoring Undergraduate Research: A Personal Journey
Abstract: Over the last 15 years I have mentored about a dozen different summer undergraduate research projects in applied dynamical systems. Ten of the projects have led to publications, and of these three were in peer-reviewed journals. In my talk I will touch upon the problems the students looked at (shallow water waves over rough surfaces, eigenvalues of matrix pencils, lead propagation in the human body, opinion formation), the results they achieved, and some of the mathematics they needed to develop and refine as they did their work. I will close by discussing in more detail the most recent project, which is concerned with modeling and analyzing a nonlinear compartment model for opinion propagation in a closed society.
June 3rd, 2024
Miguel Onorato (Department of Physics, University of Turin)
Title: One-Dimensional Chains to an Absorbing Surface Gravity Wave Device
Abstract: One-dimensional chains, such as the well-known Fermi-Pasta-Ulam-Tsingou (FPUT) chain, have long served as crucial models for exploring the fundamental properties of nonlinear and dispersive systems. In this presentation, I will begin by examining a modified version of the FPUT model that includes a resonator, i.e., an additional spring and mass attached to the main chain's masses. This configuration is sometimes referred to as the Kelvin chain in the literature. A notable feature of this chain is that it exhibits two or more branches in the dispersion relation, depending on the resonators' positions. This model has inspired the development of a device named MetaReef, designed to address the issue of beach erosion by attenuating surface gravity waves. Following an introduction to the Kelvin chain, I will present experimental research conducted in a wave flume to evaluate MetaReef's capabilities.