SCIENTIFIC PROGRAMS AND ACTIVITIES

November 28, 2014

April 28-29, 2006 Young Mathematicians' Conference in PDE and Dynamical Systems III

Organizers: Walter Craig , Adrian Nachman, Dmitry Pelinovsky, Mary Pugh , Catherine Sulem

Supported by
 To register to attend email gensci(PUT_AT_SIGN_HERE)fields.utoronto.ca Young Mathematicians' Conference 2004 Hotels and Housing Fields Visitor Information Young Mathematicians' Conference 2005 Audio and Slides

A conference of junior researchers in the areas of PDE and Dynamical Systems. will be held April 28-29 at the Fields Institute , Toronto.The goal is to encourage scientific exchange, and to create an opportunity for mathematicians in an early stage of their career to get to know each other and each other's work.

 Friday April 28 Session Chair: Mary Pugh 9:30-10:30 Lecture: Almut Burchard (University of Toronto) On an isoperimetric conjecture for a Schrodinger operator depending on the curvature of a loop 10:30-11:00 Allen Tesdall (University of Houston) The triple point paradox for the nonlinear wave system 11:00-11:30 coffee break 11:30-12 Kristian Bjerklov (University of Toronto) Schroedinger equations with quasi-periodic potentials 12:00-12:30 Wenxiang Liu (University of Alberta) A Mathematical Model for Cancer Treatment by Cell Cycle-Specific Chemotherapy 12:30-2:30 lunch break Session Chair: Walter Craig 2:30-3:00 Zoi Rapti (University of Illinois at Urbana-Champaign) Instability for Nonlinear Schroedinger Equations with a Periodic Potential 3:00-3:30 Yujin Guo (University of British Columbia) Partial differential equations arising from electrostatic MEMS 3:30-4:00 Ivana Alexandrova, University of Toronto The Scattering Amplitude at a Maximum of the Potential 4:00-4:30 coffee break 4:30-5:00 Marina Chugunova (McMaster University) Existence and stability of two-pulse solutions in the fifth-order KdV equation 5:00-5:30 Steven Dejak (University of Toronto) Long-Time Dynamics of KdV Solitary Waves over a Variable Bottom 5:30-6:00 Jianfu Ma, York University Multistability in Spiking Neuron Models of Delayed Recurrent Inhibitory Loops Saturday April 29 Session Chair: Catherine Sulem 9:30-10:30 Lecture: Dmitry Pelinovsky (McMaster University) Traveling waves in nonlinear lattices 10:30-11:00 Roy Wilds (McGill University) A new approach for analyzing the solution structure of traveling waves in lattice differential equations 11:00-11:30 coffee break 11:30-12:00 Gergely Röst (York University) Bifurcation of periodic equations with delay 12:00-12:30 Patrick Boily (University of Ottawa) Spiral Wave Anchoring Under Full Euclidean Symmetry-Breaking 12:30-2:00 lunch break Session Chair: Adrian Nachman 2:00-2:30 Elaine Spiller (University at Buffalo) The dispersion-managed nonlinear Schroedinger equation: symmetries, linearized modes, and applications 2:30-3:00 Slim Ibrahim (McMaster University) Suitable weak solutions of The Navier-Stokes system 3:00-3:30 coffee break 3:30-4:00 Gang Zhou (University of Toronto) Formation of singularities of reaction diffusion equations 4:00-4:30 Nikos Tzirakis (University of Toronto) Improved global well-posedness for the Zakharov and the Klein-Gordon-Schrodinger systems 4:30-5:00 Jackson Chan (University of Toronto) Positivity of Lyapunov exponent of quasi-periodic Schrodinger equation

ABSTRACTS

Ivana Alexandrova, University of Toronto
The Scattering Amplitude at a Maximum of the Potential
We consider the semi-classical scattering amplitude for short range potential perturbations of the Laplacian at an energy level which is a maximum of the potential. We prove that the scattering amplitude quantizes the (appropriately defined) scattering relation in the sense of semi-classical Fourier integral operators.
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Kristian Bjerklov (University of Toronto)
Schroedinger equations with quasi-periodic potentials.

We present some dynamical and spectral results about the quasi-periodic Schroedinger equation. We will emphasise on a
dynamical systems approach for the investigation, and objects like Strange Nonchaotic Attractors will be introduced and studied.

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Patrick Boily (University of Ottawa)
Spiral Wave Anchoring Under Full Euclidean Symmetry-Breaking

The spiral is one of Nature's more ubiquitous shape: it can be seen in various media, from galactic geometry to cardiac tissue. In the literature, very specific models are used to explain some of the observed incarnations of these dynamic entities. Barkley first noticed that the range of possible spiral behaviour is caused by the Euclidean symmetry that these models possess. In experiments however, the physical domain is never perfectly Euclidean. The heart, for instance, is finite, anisotropic and littered with inhomogeneities. To capture this loss of symmetry and as a result model the physical situation with a higher degree of accuracy, LeBlanc and Wulff introduced forced Euclidean symmetry-breaking (FESB) in the analysis, via two basic types of perturbations: translational symmetry-breaking (TSB) and rotational symmetry-breaking terms. They show that phenomena such as anchoring and quasi-periodic meandering can be explained by combining Barkley's insight with FESB. In this talk, we provide a characterization of spiral anchoring by studying the effects of full FESB, combining RSB terms with simultaneous TSB terms.
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Almut Burchard (University of Toronto)

On an isoperimetric conjecture for a Schrodinger operator depending on the curvature of a loop.

Let C be a smooth closed curve of length 2 Pi in three dimensionals, and let k(s) be its curvature, regarded as a function of arclength. This curve determines the one-dimensional Schrodinger operator H_C=-d^2/ds^2 + k^2(s) acting on the space of square integrable 2 Pi - periodic functions. A natural conjecture is that the lowest spectral value e(C) is bounded below by 1 for any curve (the value is assumed when C is a circle). In recent joint work with L. E. Thomas we study a family of curves that includes the circle and for which e(C)=1 as well. We show that the curves in this family are local minimizers, i.e., e(C) increases under small perturbations leading away from the family. In the talk, I will explain our interest in the problem, describe how such Schrodinger operators appear in various problems in Mathematical Physics, and sketch the proof of our result.

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Wenxiang Liu (University of Alberta)

A Mathematical Model for Cancer Treatment by Cell Cycle-Specific Chemotherapy

In this paper we use a mathematical model to study the effect of a cell-cycle specific drug on the development of cancer, including the immune response. The cancer cells are split into the mitotic phase (M-phase), the quiescent phase (G0-phase) and the interphase (G1; S; G2 phases). We include a time delay for the passage through the interphase. The immune cells interact with all cells and the drug is assumed to be M-phase specific. We study analytically and numerically the stability of the cancer-free equilibrium and we show that the M-phase specific drug does not change its stability. Nevertheless, the M-phase drug significantly reduces cancer growth. Moreover we find oscillations through a Hopf bifurcation. Finally, we use the model to discuss the efficiency of cell synchronization before treatment (synchronization method).

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Jackson Chan (University of Toronto)
Positivity of Lyapunov exponent of quasi-periodic Schrodinger equation

Positivity of Lyapunov exponent is essential in proving that the eigenfunction of the discrete Schrodinger equation decays exponentially. In this talk, I will define a notion of "typical" potential, and show that for "typical" C^3 potential, the Lyapunov exponent is positive for all energies.

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Marina Chugunova (McMaster University)

Existence and stability of two-pulse solutions in the fifth-order KdV equation

I will discuss application of Lyapunov-Schmidt reductions method to construct two-pulse solutions in the fifth order Korteweg-de Vries equation. Stablility of the two-pulse solutions is investigated numerically. It turns out that one half of the two-pulse solutions is stable and one half is unstable.

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Steven Dejak (University of Toronto)
Long-Time Dynamics of KdV Solitary Waves over a Variable Bottom

We study the variable bottom generalized Korteweg-de Vries (bKdV) equation p_t u=-p_x(p_x^2 u+f(u)-b(t,x)u), where f is a nonlinearity and b is a small, bounded and slowly varying function related to the varying depth of a channel of water. Many variable coefficient KdV-type equations, including the variable coefficient, variable bottom KdV equation, can be rescaled into the bKdV. We study the long time behaviour of solutions with initial conditions close to a stable, b=0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave, whose centre and scale evolve according to a certain dynamical law involving the function b(t,x), plus an H^1-small fluctuation.

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Yujin Guo (University of British Columbia)

Partial differential equations arising from electrostatic MEMS

We analyze the nonlinear parabolic problem $u_t= \Delta u - \frac{\lambda f(x)}{(1+u)^2}$ on a bounded domain $\Omega$ of $R^N$ with Dirichlet boundary conditions. This equation models the dynamic deflection of a simple electrostatic Micro-Electromechanical System (MEMS) device, consisting of a thin dielectric elastic membrane with boundary supported at $0$ above a rigid ground plate located at $-1$. Here $f(x) \geq 0$ characterizes the varying dielectric permittivity profile. When a voltage --represented here by $\lambda$-- is applied, the membrane deflects towards the ground plate and a snap-through (touchdown) may occur when it exceeds a certain critical value $\lambda ^*$. Applying analytical and numerical techniques, the existence of $\lambda ^*$ is established together with rigorous bounds. We show the existence of at least one steady-state when $\lambda < \lambda ^*$ (and when $\lambda =\lambda ^*$ in dimension N < 8), while none is possible for $\lambda >\lambda ^*$. More refined properties of steady states, such as regularity, stability, uniqueness, multiplicity, energy estimates and compactness results, are shown to depend on the dimension of the ambient space and on the permittivity profile. For the dynamic case, the membrane globally converges to its unique maximal steady-state when $\lambda \leq \lambda ^*$; on the other hand, if $\lambda >\lambda ^*$ the membrane must touchdown at finite time, and touchdown can not take place at the location where the permittivity profile vanishes. This is joint work with Nassif Ghoussoub at UBC.

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Slim Ibrahim (McMaster University)

On suitable weak solutions of The Navier-Stokes system

Weak solutions of the Navier -Stokes equation are suitable if they satisfy a localized version of the energy inequality. The interest. In this notion is that the partial regularity theorems of Scheffer, and Caffarelli, Kohn and Nirenberg apply to suitable weak solutions of the Navier-Stokes equations in three spatial dimensions, limiting the parabolic Hausdorff dimension of their singular set.

In this paper we discuss the class of suitable weak solutions and some of its properties. We show that the weak solutions obtained by the approximation method of Leray are suitable, as are weak solutions obtained by the super-viscosity approximation. However it is not known whether the weak solutions obtained by Hopf's method of Galerkin approximation are suitable. For the problem on the 3D torus we give a new estimate of weak solutions which has some bearing on this question.

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Jianfu Ma (York University)
Co-author Jianhong Wu
Multistability in Spiking Neuron Models of Delayed Recurrent Inhibitory Loops
We show that spiking neuron models (such as the linear or quadratic integrate-and-fire model) with delayed feedback can generate a large number of asymptotically stable periodic solutions with complicated binary patterns, if more biological realities of firing and rebound as well as the absolute refractory period are incorporated.
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Demitry Pelinovsky (McMaster University)
Traveling waves in nonlinear lattices

In recent years, exceptional discretizations of nonlinear PDEs have been constructed to support translationally invariant kinks and solitary waves, i.e. families of nonlinear waves centered at arbitrary points between the lattice sites. It has been suggested that the translationally invariant stationary solutions may persist as traveling solutions for small velocities. I will explain analysis and numerical algorithms which can be used to study and test the existence of traveling wave solutions.
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Zoi Rapti (University of Illinois at Urbana-Champaign)

Instability for Nonlinear Schroedinger Equations with a Periodic Potential

In this talk, we will consider 1-dimensional Nonlinear Schrodinger Equations with a periodic potential and will study the stability properties of periodic solutions. We will show how, by exploiting the symmetries of the problem, we can develop a simple sufficient criterion that guarantees the existence of modulational instability. In the case of small amplitude solutions bifurcating from the band edges of the linear problem, we show that the lower band edges are unstable in the focusing case, while the upper band edges are unstable in the defocusing case. This is joint work with Jared C. Bronski.

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Gergely Röst (York University)
Bifurcation of periodic equations with delay

We study the bifurcation of time-periodic differential equations with delay, depending on a parameter. The complete bifurcation analysis is performed explicitly, using Floquet-multipliers, spectral projection and center manifold reduction. The results are extended to the case of strong resonance. Numerous examples are given to illustrate our
theorems: subcritical and supercritical bifurcations into invariant tori can be observed for many notable equations. Equations with diffusion are also discussed.

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Elaine Spiller (University at Buffalo)
The dispersion-managed nonlinear Schroedinger equation: symmetries, linearized modes, and applications

The nonlinear Schroedinger equation (NLS) with a periodic, varying dispersion coefficient models the dynamics of light in dispersion- managed communication systems and mode-locked lasers. The dispersion- managed nonlinear Schroedinger equation (DMNLS) is an averaged version of NLS which restores some symmetries that are lost in NLS when the dispersion coeficient is not constant. I will discuss these symmetries, the corresponding conservation laws, and modes of the linearized DMNLS. I will also discuss how these linearized modes can be utilized to guide importance-sampled Monte-Carlo simulations of rare events in dispersion-managed lightwave systems subject to noise. This study is pertinent because the performance of lightwave systems is limited by the occurrence of rare events.
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Allen Tesdall (University of Huston)

The triple point paradox for the nonlinear wave system.

Experimental observations of the reflection of weak shock waves off a thin wedge show a pattern that closely resembles Mach reflection, in which the incident, reflected, and Mach shocks meet at a triple point. However, the von Neumann theory of shock reflection shows that a triple point configuration, consisting of three shocks and a contact discontinuity meeting at a point, is impossible for sufficiently weak shocks. This apparent conflict between theory and experiment has been a long-standing puzzle, and is often referred to as the triple point, or von Neumann, paradox.

We present numerical solutions of a two-dimensional Riemann problem for the nonlinear wave system that is analogous to the reflection of weak shocks off thin wedges. The solutions contain a remarkably complex structure: there is a sequence of triple points and supersonic
patches immediately behind the leading triple point, formed by the reflection of weak shocks and expansion waves between the sonic line and the Mach shock. An expansion fan originates at each triple point, resolving the von Neumann paradox. These results are consistent
with numerical solutions of a problem for the unsteady transonic small disturbance equation that describes the Mach reflection of weak shocks, and with recent experimental evidence that shows a sequence of shocks and expansions behind the triple point in aweak shock Mach reflection.
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Nikos Tzirakis (University of Toronto)

Improved global well-posedness for the Zakharov and the Klein-Gordon-Schrodinger systems

In this talk I will prove low-regularity global well-posedness for the 1d Zakharov (Z) and 1d, 2d, and 3d Klein-Gordon- Schroedinger system (KGS), which are systems in two variables (u, n). Z is known to be locally well-posed in (u, n) \in L^{2} \times H^{-1/2} and KGS is known to be locally well-posed in (u, n) \in L^{2} \times L^{2}. I will show that Z and KGS are globally well-posed in these spaces, respectively, by using an available conservation law for the L^{2} norm of u and controlling the growth of n via the estimates in the local theory. This is joint work with Jim Colliander and Justin Holmer.

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Roy Wilds (McGill University)
A new approach for analyzing the solution structure of traveling waves in lattice differential equations

Lattice Differential Equations (LDEs) are systems of coupled ODEs. They arise naturally in a diverse range of fields, such as in modeling of biological systems and descriptions of materials at the atomic/molecular level. Traveling Waves (TWs) represent a fundamental solution class of these problems. The theory and computation of TWs in LDEs has only begun to receive a great deal of attention over the last 25 years. The analysis of such problems is quite difficult, owing to the presence of mixed-type functional differential equations (differential equations with delays and advances present). I will present a new approach we have developed which allows us to approximate the functional differential equations by a closed system of ODEs. By applying this technique to several example problems, I will show how this new formulation allows one to infer many characteristics of the traveling waves in the lattice differential equation, using standard dynamical systems analysis.

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Gang Zhou (University of Toronto)
Formation of singularities of reaction diffusion equations

The formation of singularities of reaction diffusion equations arise in the motion by mean curvature flow, vortex dynamic in superconductors and mathematical biology. Our result shows, under certain conditions on the datum, the asymptotic behavior of the solution before the time forming singularities. Moreover we give the remainder estimate.

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