Schedule for: 16w5050 - Coherent Structures in PDEs and Their Applications

A welcome drink will be served at the hotel.

Atomic-gas Bose-Einstein condensates (BECs) enable unique opportunities for laboratory studies of superfluid dynamics, such as the complex flows found in two-dimensional quantum turbulence (2DQT). Although experimental and theoretical study of turbulent flows in classical fluids is often especially challenging, 2DQT in a BEC may be readily characterized by the dynamics of quantized vortices, which can be generated, manipulated, and optically detected with laser light. Additionally, analytical and numerical methods provide predictions of vortex dynamics and 2DQT characteristics that can be directly tested in the laboratory. Our understanding of 2DQT and other superfluid phenomena thus relies upon an understanding of vortex dynamics and their effective interactions within few-vortex and many-vortex systems. In this talk, we will discuss recent research on 2DQT and vortex dynamics in BECs in order to give an overview of this field, focusing on laboratory achievements, techniques, and realistic future avenues of study, as well as how experiments may inform and are aided by ongoing theoretical research.

The chaotic motion of quantum vortices in planar Bose-Einstein condensates represents a minimal realisation of turbulent fluid dynamics, accessible in current experiments. I will outline recent theoretical progress in creating and detecting such states, in the context of both the strongly forced and statistical steady state regimes. Possible signatures of coherent vortex structures, and their role in emergent superfluid viscosity and negative temperature states will be discussed.

In a recent experiment conducted at NIST a ring Bose–Einstein condensate (BEC) was prepared in a unit angular momentum circulation state. A barrier was then slowly raised and left on for a variable hold time and then turned off. The final circulation of the BEC was studied as a function of hold time and barrier energy height. This procedure was carried out for several well– characterized non–zero temperatures. We have studied the energetics of this process under the assumption that a vortex is initially present in the center of the ring BEC and then travels out of the ring through the density notch created by the barrier. We have computed the energy per par- ticle of the condensate system for a variable location of the vortex by solving the time–dependent Generalized Gross–Pitaevskii (GGP) equation in imaginary time. To account for finite– tem- perature we solved self–consistently for the condensate fraction as a function of temperature in thermal equilibrium for fixed total particle number. This yielded the non–condensate density which appears in the GGP affecting the energy of the vortex. We also modeled the dynamics of the vortex using the ZNG formalism.

Bose-Einstein condensation, theoretically known to appear at ultralow temperatures since the 1920s, was achieved in the laboratory only 20 years ago, in dilute bosonic gases at temperatures close to absolute zero. The wide range of applications and controlabillity of the system parame- ters has made them unprecedented tools for exploring novel quantum phases and behavior. The most common theoretical method employed to tackle these complex systems, typically consisting of tens of thousands of interacting particles is the celebrated mean-field Gross-Pitaevskii model. Even though it has been proven successful in describing various types of nonlinear excitations it does not take into consideration fragmentation and correlations that can develop in time. I will briefly present a systematic theory, the MultiConfigurational Time-Dependet Hartree for Bosons (MCTDHB) [1] that has been developed in order to solve the many-body Schroedinger equation beyond the mean-field approach and can be in principle exact, and the latest numerical imple- mentation (solver) that is freely distributed in the web [2]. I will then discuss some particular applications in Bose gases possessing angular momentum and interacting vortices and discuss the novel concept of phantom vortices. The latter are topological defects in rotating gases that evade detection in the density of the gas but give their signature in the correlation function. Last I will present recent theoretical and experimental investigations in nearly-1D gases where periodic modulation of the scattering length might give rise to beyond-mean-field phenomena. [1] Alexej I. Streltsov, Ofir E. Alon, and Lorenz S. Cederbaum, Role of Excited States in the Splitting of a Trapped Interacting Bose-Einstein Condensate by a Time-Dependent Barrier, Phys. Rev. Lett. 99, 030402 (2007); Ofir E. Alon, Alexej I. Streltsov, and Lorenz S. Cederbaum, Mul- ticonfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems, Phys. Rev. A 77, 033613 (2008). [2] A.U.J. Lode, M.C. Tsatsos and E. Fasshauer, The Multiconfigurational Time-Dependent Hartree for Indistinguishable particles software (2016), http://ultracold.org.

Ever since the seminal work of Bethuel-Brezis-Helein(BBH) on a simplified 2d Ginzburg-Landau system rose to prominence in the ’80s, it has inspired a very active field in nonlinear analysis and mathematical physics that connects the theory of harmonic maps, energy renormalization and hard-analysis concentration estimates. However, up to this date, the beautiful description of vortex configurations achieved in the 2d setting is yet to find an exact analog in higher dimensions; this is due to a weaker characterization of vortex defects, the delicate interplay between the geometry of these codimension 2 objects and their mutual interaction, in dimensions greater than or equal to 3. In this talk I will present a framework that allows to capture an ”effective interaction energy” of nearly parallel vortex filaments in certain 3d settings: this provides a next order or ”renormalization” of the energy of clustering-filaments configurations together with a more accurate description of the vortex region in the spirit of the BBH study in 2d. This is joint work with Robert Jerrard.

The Gross-Pitaevskii equation for a Bose-Einstein condensate with symmetric harmonic trap is analyzed. Periodic solutions play an important role in the understanding of the long term behavior. In this talk we prove the existence of several global branches of solutions among which there are vortex solutions and dipole solutions.

Asymptotic reductions of a defocusing nonlocal nonlinear Schrdinger model in (2+1)-dimensions, in both Cartesian and cylindrical geometry, are presented. First, at an intermediate stage, a Boussinesq equation is derived, and then far-field, KadomtsevPetviashvilli I and II (KP-I, KP- II) equations for right- and left-going waves are found. This way, small-amplitude, planar or ring-shaped, dark or anti-dark solitons are derived, whose nature and stability is determined by a parameter defined by the physical parameters of the original nonlocal system. It is shown that (dark) anti-dark solitons are supported by a weak (strong) nonlocality, and are unstable (stable) against transverse perturbations. The analytical predictions are corroborated by direct numerical simulations.

We review recent results for the emergence, existence, dynamics and interactions of vortex rings in Bose-Einstein condensates (BECs) modelled by the Gross-Pitaevskii equation (GPE). We focus our attention on the two opposite regimes of low and high atomic density limits in the BEC as well as in the intermediate transition between these two limits. In the low density limit, corresponding to the linear limit, we study the emergence of single and multiple vortex rings emanating from planar 3D dark solitons through bifurcations. We characterize such bifurcations quantitatively using a Galerkin-type approach, and find good qualitative and quantitative agreement with our Bogoliubov-de Gennes (BdG) numerical analysis. Under appropriate conditions for the trapping strengths, we find that vortex rings might be stabilized for large enough atomic densities (large chemical potentials). On the other hand, in the large density limit, the vortex rings acquire stability and are effec- tively robust coherent structures. We study different single and multi-vortex-ring configurations together with their (normal) modes of vibration. Exotic structures such as Hopfions, the one- component counterpart to Skyrmions, are also constructed and tested for stability. Finally, we discuss some interactions dynamics between vortex rings such as periodic leapfrogging of co-axial vortex rings and the scattering behavior for co-planar collisions between vortex rings. Work in collaboration with: P.G. Kevrekidis, D.J. Frantzeskakis, Wenlong Wang, R.M. Caplan, J.D. Talley, R.N. Bisset, C. Ticknor, and L.A. Collins.

Motivated by work in nonlinear optics, as well as more recently in Bose-Einstein condensate mix- tures, we will explore a series of nonlinear states that arise in such systems. We will start from a single structure, the so-called dark-bright solitary wave, and then expand our considerations to multiple such waves, their spectral properties, nonlinear interactions and experimental observa- tions. A twist will be to consider the dark solitons of the one component as effective potentials that will trap the bright waves of the second component, an approach that will also prove useful in characterizing the bifurcations and instabilities of the system. Beating so-called dark-dark soliton variants of such states will also be touched upon. Generalizations of all these notions in higher dimensions and, so-called, vortex-bright solitons will also be offered and challenges for future work will be discussed.

Swarming patterns arising from self-propelled particles have been extensively studied particu- larly in two-dimensions and in the absence of an embedding medium. We consider the dynamics of more realistic three dimensional self-propelled particles interacting in a fluid medium. The fluid interaction terms generated by direct short-ranged pairwise interactions may impart much longer-ranged hydrodynamic forces, effectively amplifying the coupling between individuals. We study two limiting cases of fluid interactions, a ”clear fluid” where particles have direct knowl- edge of their own velocity, that of others and of the fluid, and an ”opaque fluid” where particles are able to determine their velocity only in relation to the surrounding fluid flow. We discover a number of new collective three-dimensional patterns including flocks with prolate or oblate shapes, recirculating peloton-like structures, and jet-like fluid flows that entrain particles medi- ating their escape from the center of mill-like structures. We also discuss how fluid flows may stabilize emergent patterns that would be short-lived in fluid free environments.

Mesenchymal migration refers to a proteolytic and path generating strategy of individual cell motion. In mesenchymal migration, through the action of proteases, degradation and remodeling of the extracellular matrix of the surrounding tissue is executed, creating tube-like matrix defects along the path of migration. The extracellular matrix (ECM) is a 3D fibre network composed primarily of collagen; during cell migration provides directional information by the orientation of the fibres, this process is known as contact guidance. In 2006, Hillen put forward the $M^5$-Model model for mesenchymal migration, which takes into account the interaction between migrating cells and the ECM. The model consists of two coupled equations: a transport equation for the population of cells and a integro-differential equation for the dynamic of the matrix. Afterwards, Wang et al. proved the existence of families of standing and traveling wave solutions in the one-dimensional version of the model. The families are composed of pulses for the cells and decreasing fronts for the matrix. These families are indexed by the wave speed and their members have different amplitudes among them. Our study depart from their results. We have found that all the members of the family of standing waves are spectrally stable regardless of their amplitude. We first show that the essential spectrum is a subset of the left-half complex plane. Later, using energy estimates we prove that the point spectrum is empty. This colud be the first step into the way of proving nonlinear stability.

A wide variety of physical and biological systems can be described as continuum limits of inter- acting particles. Their dynamics can often be described in terms of a monotonically decreasing interaction energy that is often non-local in nature. We show how to exploit these energies nu- merically, analytically and asymptotically to characterize the observed behavior. Examples are drawn from the dynamics of thin fluid layers. We show how to use these ideas to derive a simple characterization of the convoluted fingered domains observed experimentally in ferrofluids. Joint work with Jaron Kent-Dobias (HMC 2014), Department of Physics Cornell University

Biological aggregations such as insect swarms and fish schools may arise from a combination of social interactions and environmental cues. Nonlocal continuum equations are often used to model aggregations, which manifest as localized solutions. While popular in the literature, the nonlocal models pose significant analytical and computational challenges. Beginning with the nonlocal aggregation model of [Topaz et al., Bull. Math. Bio., 2006], we derive the minimal well- posed long-wave approximation, which is a degenerate Cahn-Hilliard equation. Using analysis and computation, we study energy minimizers and show that they retain many salient features of those of the nonlocal model. Furthermore, using the Cahn-Hilliard model as a testbed, we investigate how an external potential modeling food sources can suppress peak population density, which is essential for controlling locust outbreaks. Random potentials tend to increase peak density, whereas periodic potentials can suppress it. Joint work with Andrew Bernoff.

In this talk, we explore some of the various issues that may occur in attempting to fit a dynamical systems (either agent- or continuum-based) model of urban crime to data on just the attack times and locations. We show how one may carry out a regression analysis for the model described by [M.B. Short, et al., Math. Mod. Meth. Appl. Sci. 2008] by using simulated attack data from the agent-based model. It is discussed how one can incorporate the attack data into the partial differential equations for the expected attractiveness to burgle and the criminal density to predict crime rates between attacks. Using this predicted crime rate, we derive a likelihood function that one can maximise in order to fit parameters and/or initial conditions for the model. Finally, we outline future research in this area where we believe that the combination of dynamical systems modelling, analysis, and data assimilation can prove effective in developing policing strategies for urban crime.

We consider a first-order aggregation model in both discrete and continuum formulations and show how it can be obtained as zero inertia limits of second-order models. The limiting procedure becomes particularly important when one considers anisotropy in the first-order discrete model, as in that case the model becomes {\em implicit}, and issues such as non-uniqueness and jump discontinuities are being brought up. To extend solutions beyond breakdown we propose a relaxation system containing a small parameter \(\epsilon\), which can be interpreted as a small amount of inertia or response time. We show that the limit \(\epsilon \to 0\) can be used as a jump criterion to select the physically correct velocities. In the continuum setting, the procedure consists in a macroscopic limit, enabling the passage from a kinetic model for aggregation to an evolution equation for the macroscopic density. This is joint work with Joep Evers, Lenya Ryzhik and Weiran Sun.

We consider a system of two aggregation equations. This system consists of two continuity equations for the densities \(\rho_1\) and \(\rho_2\), coupled via the the velocities \(v_1\) and \(v_2\). Each \(v_i\) is given by \(v_i = -\nabla K_{ii} * \rho_i -\nabla K_{ij} * \rho_j\), where the former convolution term accounts for self-interactions and the latter one for cross-interactions. Each kernel is chosen such that the interactions exhibit Newtonian repulsion and linear attraction (i.e., the kernel is quadratic). The free parameters in the model are the coefficients multiplying the repulsion and attraction parts. We assume that the interactions are symmetric, in the sense that \(K_{11}=K_{22}\) and \(K_{12}=K_{21}\). Our aim is to characterise the steady states of the model and their stability for varying model parameters. For our specific interaction kernels, it is known that the density can take only two different values in a steady state, depending on whether the two densities coexist at a certain position or not. However, the geometry of the supports of \(\rho_1\) and \(\rho_2\) is far from trivial.

The inclusion of cross-diffusion in a model is important when trying to understand the dynamics of a multi-species population where the populations try to avoid each other. Such models have been proposed since the 50’s and its effects have been analyzed vastly in the literature. Cross- diffusion can lead to many interesting patterns and generally makes it much harder to answer fundamental questions, such as the global well-posedness and regularity of the system. We discuss a system that was derived as a particle model for population segregation. Previous numerical results have demonstrated both strong and weak-segregation. In this talk, I begin by deriving the continuum model based on Certain rules of engagement for interacting particles. I will also discuss the existence of steady-state solutions and mention the effect that the environment and various mobilities (how easily a population can navigate the environment) has on the qualitative behavior of the solution. Furthermore, we discuss the global well-posedness for the system pointing out the main difficulties.

Significant progress has been made in recent years on understanding bulk pattern formation in large particles systems. In this talk we will review the dynamical systems approach to under- standing how particles with isotropic interaction over varying length scales assemble into global patterns. We then leverage this theory to provide physical insight into the nature of spherical assembly problems. Time permitting we will preview new theoretical results that extend this pattern formation to the challenging non-isotropic setting.

Motivated by earlier studies of artificial perceptions of light called phosphenes, we analyze trav- eling wave solutions in a chain of periodically forced coupled nonlinear oscillators modeling this phenomenon. We examine the discrete model problem in its co-traveling frame and system- atically obtain the corresponding traveling waves in one spatial dimension. Direct numerical simulations as well as linear stability analysis are employed to reveal the parameter regions where the traveling waves are stable, and these waves are, in turn, connected to the standing waves analyzed in earlier work. We also consider a two-dimensional extension of the model and demonstrate the robust evolution and stability of planar fronts and annihilation of radial ones. Finally, we show that solutions that initially feature two symmetric fronts with bulged centers evolve in qualitative agreement with experimental observations of phosphenes.

In this talk I will discuss the problem of two elastic beams deflecting in an electric field. The beams are held opposite so that they will come into physical contact, if the electric field is strong enough. This contact is manifested as a singularity in the governing PDEs. We introduce a regularized model which is globally well-posed and permits dynamics through the singularity. The post contact dynamics are analyzed with numerical and perturbation methods and detailed interfacial dynamical laws are established and validated. We find new stable equilibrium states which raises the potential for bistability in the system. Estimates on the bistable parameter range are found by means of detailed singular perturbation analysis. This is joint work with Joceline Lega (Arizona) and Karl Glasner (Arizona).

In this talk I will construct and analyze a model of cell receptor aggregation. We study the impact of density dependent diffusion on aggregate formation in a one-dimensional domain. Critical values of receptor diffusion and receptor activation are found and compared with numerical simulations. In the case of receptor activation the analytical results agree very well with the numerical calculations. Finally, we consider our model in higher dimensional domains. In this case our analysis is primarily numerical.

We formulate and analyze a class of coupled cell-bulk PDE models in 2-D bounded domains. Our class of models, related to the study of quorum sensing, consists of m small cells with multi- component intracellular dynamics that are coupled together by a diffusion field that undergoes constant bulk decay. We assume that the cells can release a specific signaling molecule into the bulk region exterior to the cells, and that this secretion is regulated by both the extracellular con- centration of the molecule together with its number density inside the cells. By first constructing the steady-state solution, and then studying the associated linear stability problem, we show for several specific cell kinetics that the communication between the small cells through the diffusive medium leads, in certain parameter regimes, to the triggering of synchronized oscillations that otherwise would not be present in the absence of any cell-bulk coupling. Moreover, in the well- mixed limit of very large bulk diffusion, we show that the coupled cell-bulk PDE-ODE model can be reduced to a finite dimensional system of nonlinear ODEs with global coupling, that exhibits quorum-sensing behavior. The analytical and numerical study of these limiting ODEs reveals the existence of globally stable time-periodic solution branches that are intrinsically due to the cell-bulk coupling. Joint with Jia Gou (UBC) and Michael ward (UBC)

I will present a Hamiltonian mean-field model. The model provides a simplified description of transport in marginally stable systems including vorticity mixing in strong shear flow and electron dynamics in plasmas. Self-consistency is incorporated through a mean-field that couples all the degrees of freedom. The model is formulated as a large set of N coupled standard-like twist maps. Invariant tori and their breakup play a central role in the study of global transport in these self-consistent map examples. I will present an algorithm to compute, continue and approximate the breakdown of analyticity of invariant tori in a simplified version of a self-consistent model. This is joint work with Diego del Castillo, David Martinez and Arturo Olvera.

We develop a variational method of deriving stochastic partial differential equations whose so- lutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa- Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to stochastic partial differential equations (SPDE). In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler-Poincar ́e structure of the CH equation, and it also does not occur for the original peakon solutions of the unperturbed deterministic CH equa- tion. The discussion raises issues about the science of stochastic deformations of evolutionary PDE and the sensitivity of the resulting solutions of the SPDE to the choices made in stochastic modelling.

Optical graphene, or an optical honeycomb waveguide, has become a material of much interest and excitement in the optics community. This is due to the presence of Dirac points in the dispersion relationship which is a result of the symmetry of the lattice. In this talk, we study two classes of perturbations which have significant impacts on the Dirac points. The first class are so called parity-time (PT) symmetric perturbations. We show that certain types of PT- perturbations separate Dirac points while keeping the associated dispersion relationships real. This allows for novel nonlinear wave equations to be derived which model the propagation of waves in the gap. We then briefly study one-dimensional gap solitons, which are shown to be unstable. The second class of perturbations are due to helical variations in the index of refraction of the underlying optical lattice. This allows for the formation of so-called topological edge modes. In the linear regime, we find families of these modes for different classes of lattice perturbations. We then study the impact of nonlinearity on these modes showing that nonlinear edge modes form. We show via numerical experiments that these modes appear to posses the same type of stability to backscattering that the linear modes do, hinting at the existence of nonlinear topological edge-modes.

The nonlinear-Schrodinger equation admits a Hamiltonian formulation, as do many finite- dimensional reduced systems based upon it. We show some examples where methods of Hamil- tonian reduction allow us to discover and explain features in the dynamics. We apply these ideas to a few problems in nonlinear waveguides and Bose-Einstein condensates.

We present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of Fermi-Pasta-Ulam (FPU) type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations. The main feature of the FPU lattices we consider is that the number of interacting neighbors varies from site to site, and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem. This property is seen in 1-D models, and in a 3-D model with a geometry obtained from protein data. The spectral analysis of these examples suggests some non-resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic Birkhoff normal forms of the FPU systems. The invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way.

We study properties of some breather type solutions of a nonlocal discrete NLS equation mod- eling propagation in waveguide arrays built from a nematic liquid crystal substratum. The nonlocality leads to some new effects, such a internal modes in orbitally stable breathers, and nonmonotonic profiles in interfaces. We present some new theoretical results explaining some of these numerically observed features. We also discuss symmetry, monotonicity, and spectral properties of energy minimizers, and some spectal properties of shelf-type solutions.

Interfaces are considered in the slow nonlinear diffusion equation subject to a strong absorption rate. Local self-similar solutions are constructed for reversing (and anti-reversing) interfaces, where an initially advancing (receding) interface gives way to a receding (advancing) one. We use an approach based on invariant manifolds, which allows us to determine the required asymp- totic behaviour for small and large values of the concentration. We then connect the requisite asymptotic behaviours using a robust and accurate numerical scheme. By doing so, we are able to furnish a rich set of self-similar solutions for both reversing and anti-reversing interfaces. Bi- furcation diagrams are obtained and explained by reducing the linearized operator to a quantum harmonic oscillator in multi dimensions.

In the 70s, A.S Davidov proposed a discrete model to describe the transference of energy in protein chains, in which the protein chain deforms as an electron is transferred along the chain. In the continuum limit, this models develops as a coupled NLS-Boussinessq system of partial differential equations. Considering a moving reference frame, this coupled system becomes a KdV-NLS coupled system of equations. Taking advantage of the integrability of the KdV and the NLS equations by themselves, we look for traveling wave solutions of the coupled KdV-NLS system. We find several kinds of solutions for the system. This is a joint work with L. Cisneros-Ake from the Instituto Politecnico Nacional, Mexico.

We analyze examples of delayed bifurcations in reaction-diffusion systems in both the weakly and fully nonlinear regimes. The delay effect results as the system passes slowly from a stable to unstable regime, and was previously analyzed in ODEs in [P.Mandel and T.Erneux, J.Stat.Phys 48(5-6) pp.1059-1070, 1987]. For spike solutions in the fully nonlinear regime, we demonstrate that delay can be quantified for a special class of problems in which the linear stability problem is explicitly solvable. In the weakly nonlinear regime, in the context of a simplified Klausmeier model for vegetation patterns, we analyze how addition of random noise can affect the magnitude of delay. In both regimes, we show that delay can play a critical role in determining the eventual fate of the system. Joint works with Yuxin Chen, Chunyi Gai, Theodore Kolokonikov, and Michael Ward.

We consider a single spot solution for the Schnakenburg Model in a two-dimensional unit disk in the singularly perturbed limit of a small diffusivity ratio. For large values of the reaction- time constant, this spot can undergo two different types of instabilities, both due to a Hopf bifurcation. The first type induces oscillatory instability in the height of the spot. The second type induces a periodic motion of the spot center. We use formal asymptotics to investigate when these instabilities are triggered, and which one dominates. In the parameter regime where spot motion occurs, we construct a periodic solution consisting of a rotating spot, and compute its radius of rotation and angular velocity.

We consider a standard reaction-diffusion system (the Schnakenberg model) that generates lo- calized spike patterns. Our goal is to characterize the distribution of spikes and their heights in the limit of many spikes, in the presence of spatially-dependent feed rate A(x). This leads to an unusual nonlocal problem for spike locations and their heights. A key feature of the resulting nonlocal problem is that it is necessary to estimate the difference between the continuum limit and the discrete algebraic system to derive the effective spike density. In a certain limit, we find that the effective spike density scales like \(A^{2/3}(x)\) whereas the spike heights scale like \(A^{1/3}(x)\). In another limit, we derive instability thresholds for when N spikes become unstable.