
SCIENTIFIC PROGRAMS AND ACTIVITIES 

August 6, 2024  
A conference of junior researchers in the areas of PDE and Dynamical
Systems. will be held April 2829 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. ABSTRACTSIvana Alexandrova, University of Toronto
Kristian Bjerklov (University of Toronto) We present some dynamical and spectral results about the quasiperiodic
Schroedinger equation. We will emphasise on a  Patrick Boily (University of Ottawa) 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 symmetrybreaking (FESB) in the
analysis, via two basic types of perturbations: translational symmetrybreaking
(TSB) and rotational symmetrybreaking terms. They show that phenomena
such as anchoring and quasiperiodic 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. 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 onedimensional 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.  Wenxiang Liu (University of Alberta) A Mathematical Model for Cancer Treatment by Cell CycleSpecific Chemotherapy In this paper we use a mathematical model to study the effect of a
cellcycle specific drug on the development of cancer, including the
immune response. The cancer cells are split into the mitotic phase (Mphase),
the quiescent phase (G0phase) 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 Mphase
specific. We study analytically and numerically the stability of the
cancerfree equilibrium and we show that the Mphase specific drug does
not change its stability. Nevertheless, the Mphase 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).  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.  Marina Chugunova (McMaster University) Existence and stability of twopulse solutions in the fifthorder KdV equation I will discuss application of LyapunovSchmidt reductions method to construct twopulse solutions in the fifth order Kortewegde Vries equation. Stablility of the twopulse solutions is investigated numerically. It turns out that one half of the twopulse solutions is stable and one half is unstable.  We study the variable bottom generalized Kortewegde 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 KdVtype 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^1small fluctuation.  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 MicroElectromechanical 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 snapthrough (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 steadystate 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 steadystate 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.  Slim Ibrahim (McMaster University) On suitable weak solutions of The NavierStokes 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 NavierStokes 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 superviscosity 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.  Demitry Pelinovsky (McMaster University) 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. Zoi Rapti (University of Illinois at UrbanaChampaign) Instability for Nonlinear Schroedinger Equations with a Periodic Potential In this talk, we will consider 1dimensional 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.  Gergely Röst (York University) We study the bifurcation of timeperiodic differential equations with
delay, depending on a parameter. The complete bifurcation analysis is
performed explicitly, using Floquetmultipliers, spectral projection
and center manifold reduction. The results are extended to the case
of strong resonance. Numerous examples are given to illustrate our  Elaine Spiller (University at Buffalo) The nonlinear Schroedinger equation (NLS) with a periodic, varying
dispersion coefficient models the dynamics of light in dispersion managed
communication systems and modelocked 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 importancesampled
MonteCarlo simulations of rare events in dispersionmanaged lightwave
systems subject to noise. This study is pertinent because the performance
of lightwave systems is limited by the occurrence of rare events. 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 longstanding puzzle, and is often referred to as the triple point, or von Neumann, paradox. We present numerical solutions of a twodimensional 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 Nikos Tzirakis (University of Toronto) Improved global wellposedness for the Zakharov and the KleinGordonSchrodinger systems In this talk I will prove lowregularity global wellposedness for the 1d Zakharov (Z) and 1d, 2d, and 3d KleinGordon Schroedinger system (KGS), which are systems in two variables (u, n). Z is known to be locally wellposed in (u, n) \in L^{2} \times H^{1/2} and KGS is known to be locally wellposed in (u, n) \in L^{2} \times L^{2}. I will show that Z and KGS are globally wellposed 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.  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 mixedtype 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.  Gang Zhou (University of Toronto) 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. 

