In this talk, we will present an overview of recent theoretical, numerical and experimental work concerning the static, stability, bifurcation and dynamic properties of coherent structures that can emerge in one-and higher-dimensional settings within Bose-Einstein condensates at the coldest temperatures in the universe (i.e., at the nanoKelvin scale). We will discuss how this ultracold quantum mechanical setting can be approximated at a mean-field level by a deterministic PDE of the nonlinear Schrodinger type and what the fundamental nonlinear waves of the latter are, such as dark solitons and vortices. Then, we will try to go to a further layer of simplified description via nonlinear ODEs encompassing the dynamics of the waves within the traps that confine them, and the interactions between them. Finally, we will attempt to compare the analytical and numerical implementation of these reduced descriptions to recent experimental results and speculate towards a number of interesting future directions within this field
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A realistic molecular-level description of reaction-diffusion processes is often provided by spatially-discrete stochastic interacting particle systems (IPS). Spatially discreteness may be innate to the system (e.g., for reactions on 2D crystalline catalyst surfaces), or may just be a convenient modeling tool (e.g., for reactions in 1D nanoporous materials). Spatially homogeneous and heterogeneous state in these IPS can be described exactly by hierarchical master equations. These reduce to lattice differential equations, i.e., to discrete reaction-diffusion equations (RDE), after making a hierarchical truncation approximation, and the discrete RDE become continuum RDE in the hydrodynamic limit of rapid diffusion. However, the correct description of diffusion is invariably non-trivial (with concentration-dependent and tensorial diffusion coefficients), and not captured by simple approximations. Examples are provided for simple models for reaction fronts in bistable reactions such as CO-oxidation on 2D surfaces, and for catalytic conversion inside 1D linear nanopores.
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This talk is concerned with a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. The first main result is on the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically self-similar both in the area expanding and the area shriknking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. Thus the renormalized curve converges to some profile in each of the three cases, but the proof of the convergence is totally different among the three cases. This is joint work with Jong-Shenq Guo, Masahiko Shimojo and Chang-Hong Wu.
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Abstract: This lecture will focus on the legacy of Lyapunov in the area of Dichotomies. While the use of the Lyapunov-Perron theory of exponential dichotomies in the study of ODEs and PDEs is well understood, there is a potential for more insight to be gained by combining the Lyapunov-Perron theory with the related recent work of others in the area of Lyapunov Exponents and the Multiplicative Ergodic Theorem. As we will show, there is much to be gained on problems where both theories are applicable.
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Abstract: Modeling problems in environmental and physical sciences naturally leads to consider equations with discontinuous coefficients In this talk I will survey results obtained with colleagues at Oregon State University on the large scale effect on quantities of natural interest by the interface conditions required at the location of the discontinuities of the coefficients in the model. Examples that motivate this study include breakthrough curves in hydrology, occupation time in ecology and costal upwelling in oceanography. The analysis of these problems illustrates and builds on the profound connections between stochastic processes and PDE's.
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Abstract: Optimal filtering problems are important in signal processing and related fields. Most of the time the optimal filter does not admit a closed-form expression, and various numerical methods are developed to solve the filtering problem. Among these methods, particle filter is a widely used tool for numerical prediction of complex systems when observation data are available. In this talk I will first give a brief introduction on particle filtering and present error analysis on the numerical formulation of particle filter. Then I will introduce an implicit filtering method, along with its convergence results.
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Abstract: We study a sixth order convective Cahn-Hilliard type equation that describes the faceting of a growing surface. It is considered with periodic boundary conditions. We deal with the problem in one and two dimensions. We establish the existence and uniquness of weak solutions. Our goal is study the long time behavior in the one- and two-dimensional cases. We show existence of a global attractor. The numerical simulations suggest that this result is optimal.
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Abstract: In this talk we analyze the dynamics of flows generated by a class of dissipa- tive semilinear parabolic problems when some parameters of the equation vary in a topological space. We establish abstract results and apply them to partial differen- tial equations with nonlinear boundary conditions when (i) the domain of definition of the solutions vary with respect to the action of diffeomorphisms, and (ii) when some reaction and potential terms of the equation are concentrating in a narrow strip of a portion of the boundary of the domain of the solutions. Our main goal is to discuss the continuity of the nonlinear semigroup, as well as, the upper and lower semicontinuity of the family of attractors.
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Abstract: In this talk I will present some results concerning the integral $\int_a^b f(t) dY(t)$ and the differential equation: $dx(t)=f(x(t)) dY(t)$, where $Y(t)$ is a finite dimensional Holder continuous functions. We will also expand the solution $x(t)$ in term of Volterra series of $Y(t)$. When $Y(t)$ is a fractional Brownian motion, we also expand $X(t)$ as multiple Wiener-Ito integrals of $Y$ (orthogonal expansion).
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I have been do research (1) in mathematical neuroscience (rigorous mathematical analysis of existence and stability of traveling wave solutions) and (2) in nonlinear systems of fluid dynamical equations (exact limits and decay estimates with sharp rates of $L²$-norms of global strong solutions, global weak solutions of Cauchy problems). In this lecture, I will talk about my recent results.
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Inside the analysis of equations of hydrodynamic, statistical solutions have been investigated. In fact, the individual solutions may give a detailed and too complicated picture of the fluid, while one could be interested in the behavior of some global quantity related to the fluid, where the microscopic picture is replaced by the macroscopic one. This is the statistical approach to turbulence. From the mathematical point of view, we are interested in distributions invariant for these flows. Gaussian measures of Gibbsian type are associated with some shell models of 3D turbulence and 2D turbulence (GOY and SABRA models). We prove the existence of a unique global flow for a stochastic viscous shell model with the property that these Gibbs measures are invariant for these flows.
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We show the existence of a unique stochastic flow of Sobolev diffeomorphisms for stochastic differential equations (SDEs) with bounded measurable drift coefficients. This result is counter-intuitive: The dominant 'culture' in dynamical systems is that the flow `inherits' its spatial regularity from the driving vector fields. Spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the stochastic transport equation with singular coefficients (cf. work by Kunita (1990); and Flandoli-Gubinelli-Priola (2010)). The corresponding deterministic transport equation does not in general have a solution (Ambrosio (2004)). If time permits, we will construct a Sobolev differentiable stochastic flow of diffeomorphisms for one-dimensional SDEs driven by bounded measurable diffusion coefficients. No uniqueness of solutions to the SDE is presumed!
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For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.
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In 1958, Pekka Myrberg was the first to discover that as a parameter c is varied in the iterated quadratic map, periodic orbits occur with the progression of periods k, 2k, 4k, 8k, ... for a large variety of k values. Subsequently, this phenomenon of period-doubling cascades has been seen in a large variety of parameter-dependent dynamical systems, be they iterated maps, ordinary differential equations, partial differential equations, delay differential equations, or even experimental observations. It has often been observed that cascades occur in a dynamical system as the system transitions from ordered to chaotic. In this talk, I will be presenting some recent results based on work of Alligood, Mallet-Paret, and Yorke to explain the link between chaos and cascades.
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We consider a nonlinear Stefan problem, which may be used to describe the spreading of invasive species, with the free boundary representing the invasing front. In one space dimension and in the radially symmetric case with a logistic nonlinear term, it is known that this model exhibits a spreading-vanishing dichotomy. In this talk we discuss the non-radially symmetric case. By establishing suitable regularity of the free boundary, we show that the spreading-vanishing dichotomy still holds. Moreover, when spreading happens, the normalized free boundary approaches the unit sphere as time goes to infinity, and the spreading speed is the same as in the radially symmetric case. This is joint work with Hiroshi Matano (Univ of Tokyo) and Kelei Wang (Wuhan Inst of Physics and Math).
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In this talk I will present Hamiltonian identities for elliptic PDEs and systems of PDEs. I will then show some interesting applications of these identities to problems related to solutions of some nonlinear elliptic equations in the entire space. In particular, I will use the identities to show symmetry results for solutions of stationary Schrodinger equations and stationary and traveling wave solutions of the Allen-Cahn equation.
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We discuss properties of a class of dissipative dynamical systems which display rather special long time dynamics. This class is referred to as quasi-stable systems. It turns out that this is quite large class of systems naturally occurs in nonlinear PDE models arising in wave dynamics, plasma physics, thermoelasticity of plates and gas/fluid-structure interaction models. The interest in this class of systems stems from the fact that quasi-stability inequality almost automatically implies number of desirable properties such as asymptotic smoothness, finite dimensionality of attractors, regularity of attractors, exponential attraction, etc. The notion of quasi-stability is rather natural from the point of view of long-time behavior. It pertains to decomposition of the flow into exponentially stable and compact part. This represents some sort of analogy with the "splitting" method, however, the decomposition refers to the difference of two trajectories, rather than a single trajectory.
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