


Nonlinear Partial Differential Equations / Équations aux dérivées partielles non linéaires Org: Rustum Choksi (Simon Fraser) and/et Keith Promislow (Simon Fraser)
 GEORG DOLZMANN, University of Maryland, College Park, Maryland 20742, USA
Nonconvex variational problems and minimizing Young
measures

Variational integrals modeling solidtosolid phase transformations
often fail to be weakly lower semicontinuous because the energy
densities f are not quasiconvex in the sense of Morrey. In this talk
we analyse properties of minimizing Young measures generated by
minimizing sequences for these variational integrals. We prove that
moments of order q > p exist if the integrand is sufficiently close to
the pDirichlet energy at infinity. Analogous results hold true for
solutions of systems of PDE that are close to the EulerLagrange
equations for the pDirichlet energy. A counterexample related to
the onewell problem in two dimensions shows that one cannot expect in
general L^{¥} estimates, i.e., that the support of the
minimizing Young measure is uniformly bounded.
This is joint work with Jan Kristensen (Edinburgh) and Kewei Zhang
(Sussex).
 STEPHEN GUSTAFSON, University of British Columbia, Department of Mathematics,
Vancouver, British Columbia V6T 1Z2
Asymptotic stability and completeness in the energy space
for nonlinear Schroedinger equations with small solitary waves

We study nonlinear Schroedinger equations which admit families of small
solitary wave solutions. We consider solutions which are small in the
energy space H^{1}, and decompose them into solitary wave and
dispersive wave components. We establish the asymptotic stability of
the solitary wave, and the asymptotic completeness of the dispersive
wave. This is joint work with K. Nakanishi and T.P. Tsai.
 ROBERT JERRARD, University of Toronto
Vortex fliaments in Bose Einstein condensates

I will present some theorems establishing the existence of a family of
special solutions of a nonlinear partial differential equation that
describes BoseEinstein condensates subjected to rotational forcing.
The solutions in question possess geomertic structures that are
interpreted as quantized vorterx filaments. The equation being studied
is the EulerLagrange equation for a GrossPitaevsky functional
containing a small parameter e, and the solutions constructed
are local minimizers of this energy that are in a precise sense
"close" to local minimizers of a Glimiting energy. Technical
difficulties include problems associated with the degeneracy of both
the GrossPitaevsky functional and the Glimit functional.
 NATHAN KUTZ, University of Washington, Seattle, Washington 981952420, USA
Dynamics and stability of periodic solutions of the optical
parametric oscillator equations

The stability and dynamics of a new class of periodic solutions is
investigated when a degenerate OPO system is forced by an external
pumping field with a periodic spatial profile modeled by Jacobi
elliptic functions. Both sinusoidal behavior as well as localized
hyperbolic (front and pulse) behavior can be considered in this model.
The stability and bifurcation behaviors of these transverse
electromagnetic structures are studied numerically. The periodic
solutions are shown to be stabilized by the nonlinear parametric
interaction between the pump and signal fields interacting with the
cavity diffraction, attenuation, and periodic external pumping.
Specifically, sinusoidal solutions result in robust and stable
configurations while wellseparated and more localized field structures
often undergo bifurcation to new steadystate solutions having the same
period as the external forcing. Extensive numerical simulations and
studies of the solutions are provided.
 GIOVANNI LEONI, Carnegie Mellon University, Pittsburgh Pennsylvania 15213,
USA
On optimal regularity for free boundary problems

We present some results on the optimal regularity of solutions of one
and twophase free boundary problems and of free discontinuity
problems. We are particularly interested in the analyticity of local
minimizers of the MumfordShah functional and on a conjecture of De
Giorgi.
 YI A. LI, Stevens Institute of Technology
Hamiltonian approximation of the GreenNaghdi (GN)
equations to the water wave problem

In this talk, we will show that the GN equations can be derived using
the Hamiltonian structure of the full water wave problem. Under the
assumption of shallow water, i.e. long wave length compared with
the depth of the water without restriction to the wave amplitude, we
apply the Taylor expansion of the DirichletNeumann operator to the
Hamiltonian density function for the full water wave problem. As a
consequence, we justify the fact that the Hamiltonian formulation of
the GN equations is a second order approximation to that of the full
water wave problem.
 BOB PEGO, University of Maryland College Park, College Park,
Maryland 20742, USA
Dynamic scaling, Smoluchowski ripening and Burgers turbulence

Burgers turbulence is a basic model for understanding statistics of
solutions of nonlinear PDEs. Consider the inviscid Burgers equation
with random initial velocity, assumed stationary with independent
increments and no positive jumps. A meanfield model for the shock
statistics is Smoluchowski's coagulation equation with additive
kernel. I'll describe work with Govind Menon that, combined with
results of Bertoin, leads to a rather striking and complete description
of dynamic scaling limits in both models.
 DMITRY PELINOVSKI, McMaster University
Bifurcations and stability of BEC solitons in optical
lattices

I will review existence and stability of nonlinear bound states in the
GrossPitaevskii equation with the spaceperiodic potential. This
equation describes stationary states of BoseEinstein condensates in
periodic optical lattices.
Using asymptotic multiscale expansion methods, we show that the
selffocusing nonlinearity supports bifurcations of gap solitons from
all lower band edges of a periodic potential, while the selfdefocusing
nonlinearity supports bifurcations of gap solitons from all upper band
edges. We study two branches of gap solitons from each band edge and
classify stable and unstable eigenvalues in the linearized stability
problem.
 KEITH PROMISLOW, Department of Mathematics, Michigan State University
East Lansing, Michigan 48824, USA
Bifurcation in ellipticparabolic models of membrane hydration

Membrane hydration plays a key role in the operation of proton exchange
membrane (PEM) fuel cells. Recent experimental work has demonstrated the
multiplicity of steady states and hysterisis possible in autohumidified
operation. We present a degenerate nonlocal model which possesses a
degenerate family of steady states, and show that the weak effects of
diffusion generate quasisteady traveling waves which balance water
loss to the channel against water production. We also introduce
mechanical coupling issues which lead to longtime scale voltage
oscillations.
 ARND SCHEEL, University of Minnesota, School of Mathematics, Minneapolis,
Minnesota 55455, USA
Absolute instabilities of standing pulses

We analyze instabilities of standing pulses in reactiondiffusion
systems that are caused by an absolute instability of the background
state. Specifically, we investigate Turing and oscillatory
bifurcations of the homogeneous background state and their impact on
the standing pulse. At a Turing instability, symmetric pulses emerge
that are spatially asymptotic to the bifurcating spatiallyperiodic
Turing patterns. Oscillatory instabilities of the background state
typically lead to modulated pulses that emit small wave trains. We
analyse these three bifurcations by studying the standingwave
equation: the standing pulses correspond then to homoclinic orbits to
equilibria that undergo reversible bifurcations. We use blowup
techniques to show that the relevant stable and unstable manifolds can
be continued across the bifurcation point. To analyze stability, we
construct, and extend across the essential spectrum, appropriate Evans
functions for operators with algebraically decaying coefficients.
 DANIEL SPIRN, Brown University
Dynamics and instability of vortex sheets with surface
tension

The motion of an ideal twofluid model can be described solely by the
shape of the interface. When there is no surface tension, such
equations are illposed; however, when surface tension is present, the
equations become locally wellposed. We will discuss instability and
detail the dynamics up to an escapetime.
 VITALI VOUGALTER, McMaster University, Department of Mathematics,
Hamilton, Ontario L8S 4K1
Spectra of positive and negative energies in the linearized
NLS problem

We study the spectrum of the linearized NLS equation in three and
higher dimensions, in association with the energy spectrum. We prove
that unstable eigenvalues of the linearized NLS problem are related to
negative eigenvalues of the energy spectrum, while neutrally stable
eigenvalues may have both positive and negative energies.We show how
the negative index of the problem can be reduced by going to the proper
constrained subspace.

