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Nonlinear Partial Differential Equations / Équations aux dérivées partielles non linéaires
Org: Rustum Choksi (Simon Fraser) and/et Keith Promislow (Simon Fraser)

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GEORG DOLZMANN, University of Maryland, College Park, Maryland  20742, USA
Nonconvex variational problems and minimizing Young measures

Variational integrals modeling solid-to-solid 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 p-Dirichlet energy at infinity. Analogous results hold true for solutions of systems of PDE that are close to the Euler-Lagrange equations for the p-Dirichlet energy. A counterexample related to the one-well 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 H¹, 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 Bose-Einstein 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 Euler-Lagrange equation for a Gross-Pitaevsky 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 G-limiting energy. Technical difficulties include problems associated with the degeneracy of both the Gross-Pitaevsky functional and the G-limit functional.

NATHAN KUTZ, University of Washington, Seattle, Washington  98195-2420, 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 well-separated and more localized field structures often undergo bifurcation to new steady-state 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 two-phase free boundary problems and of free discontinuity problems. We are particularly interested in the analyticity of local minimizers of the Mumford-Shah functional and on a conjecture of De Giorgi.

YI A. LI, Stevens Institute of Technology
Hamiltonian approximation of the Green-Naghdi (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 Dirichlet-Neumann 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 mean-field 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 Gross-Pitaevskii equation with the space-periodic potential. This equation describes stationary states of Bose-Einstein condensates in periodic optical lattices.

Using asymptotic multi-scale expansion methods, we show that the self-focusing nonlinearity supports bifurcations of gap solitons from all lower band edges of a periodic potential, while the self-defocusing 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 elliptic-parabolic 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 quasi-steady traveling waves which balance water loss to the channel against water production. We also introduce mechanical coupling issues which lead to long-time 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 reaction-diffusion 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 spatially-periodic 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 standing-wave equation: the standing pulses correspond then to homoclinic orbits to equilibria that undergo reversible bifurcations. We use blow-up 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 two-fluid model can be described solely by the shape of the interface. When there is no surface tension, such equations are ill-posed; however, when surface tension is present, the equations become locally well-posed. We will discuss instability and detail the dynamics up to an escape-time.

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.