Title: Open Boundaries for the Nonlinear Schrödinger Equation: Numerical Analysis in Phase Space

Abstract: When solving a wave-type PDE, open boundaries must be used
to prevent spurious reflections from the computational boundary. In
this talk I introduce Phase Space Filtering, a new approach to this
problem. The Time Dependent Phase Space Filter (TDPSF) algorithm
consists of identifying and filtering outgoing waves in phase space
based on knowledge of scattering theory. It can be extended to study
wave equations in regions of phase space which are non-rectangular,
and with long range potentials.

October 10, Yi Li, Stevens Institute,

Title: Existence of Global Solutions to a System of Nonlinear Dispersive Equations

Abstract:  We investigate the nonlinear dispersive effect of
 the GN equations as second order, shallow water
 approximations to the two-dimensional full water wave problem.
 Because the GN system was derived under the assumption of long
 wave-length compared with the depth of water without restrictions
 on the wave amplitude, it contains nonlinearly dispersive terms
 and provides a good example to show nonlinearly dispersive effect
 on the dynamics of the water wave problem.

 Using a priori energy estimates and the nonlinear perturbation
 theory, we demonstrate that the GN system is locally
 well-posedness and estimate how long the GN system remains a
 good approximation to the full water wave problem.
 In addition, we demonstrate that the system possesses some
 solutions that remain in a neighborhood of certain bounded,
 oscillating functions using its Hamiltonian structure and
 comparison methods. This fact demonstrates the nonlinear
 dispersion effect on the existence of global solutions to
 the GN system as a contrast to the dispersionless, first order
 shallow water approximations. Furthermore, using
 the above results we discuss the issues of stability of solitary
 waves. Some numerical results will also be shown to illustrate
 solutions of the physical models involved.

October 29, Weizhu Bao, National University of Singapore

Title: Mathematical Analysis and Numerical Simulation of
            Bose-Einstein Condensation
Abstract:  In this talk, I review the mathematical results of the
dynamcis of Bose-Einstein condensate (BEC) and present
some efficient and stable numerical methods to compute ground states
and dynamics of BEC. As preparatory steps,
we take the 3D Gross-Pitaevskii equation (GPE) with an angular
momentum rotation, scale it to obtain a four-parameter model
and show how to reduce it to 2D GPE in certain limiting regimes.
Then we study numerically and asymptotically the
ground states, excited states and quantized vortex
states as well as their energy and chemical potential diagram
in rotating BEC. Some very interesting numerical results are
observed. Finally, we study numerically stability and interaction
of quantized vortices in rotating BEC. Some interesting interaction
patterns will be reported.  

November 15, Ricardo Barros, NJIT
Title: Two-Layer Flows with Free Surface

Abstract:
In this work we study the wave propagation in two-layer flows with free surface.
We obtain a dispersive model suited to the description of large amplitude waves propagating
in this physical system. The model is a “two-layer” generalization of the Green-
Naghdi model and can be derived by applying Hamilton’s principle to a Lagrangian
that results from the insertion of approximations directly into the Lagrangian for the
full water-wave problem. As a consequence, the variational structure of the original
problem and the corresponding symmetry properties are preserved. In addition, it is a
fully nonlinear model and deals with rotational flows. As in the case of the full problem,
the present model captures the resonance between short waves and long waves. In this
framework it is shown, by using numerical computations, the existence of homoclinic
trajectories embedded into the continuous spectrum. These correspond to true solitary
waves having the same velocities at infinity in each layer. Their study reduces to the
analysis of a Hamiltonian system with two degrees of freedom. The traveling-wave solutions
depend on three parameters: the density ratio, the depth ratio and the Froude
number based on the bottom layer. Two wave regimes, characterized by the elevation
or depression of the interface between the layers are presented. A critical depth ratio
separates these two regimes and it will be shown how it relates to a change of the
structure of the potential for the Hamiltonian system. The analysis of the number and
nature of critical points turned out to be decisive in this work. It was found that the
number of critical points can be four or two, depending on the value of the Froude
number (for fixed density and depth ratios). For sets of parameters corresponding to
oceanic conditions we have perceived the existence of true solitary waves and their
broadening whenever the wave speed increases towards a limit value. Finally, other
sets of parameters are considered for which multi-humped solitons exist, highlighting
the richness and complexity of the system considered.

November 20, Norman J. Zabusky, Weizmann Institute of Science
Title: Waves and Fluids: Simulation, Visualization and Analytics

Abstract:

I will present an overview of the role of computer simulation in
understanding and mathematizing natural phenomena at various times. I will
draw on years of work [1] beginning with the Fermi-Pasta-Ulam
near-recurrence and thermalization phenomena for a 1D nonlinear lattice
[2] and move to vortex phenomena that arise in accelerated inhomogeneous
flows, e.g., the Richtmyer-Meshkov instability in 2D and 3D [3,4]. I will
examine the choice of the:  reduced model; discrete algorithm and
visiometric representation (diagnostics), including animations.

[1] Zabusky NJ, "Fermi-Pasta-Ulam, solitons and the fabric of nonlinear
and computational science: History, synergetics, and visiometrics". CHAOS
15 (1): Art. No. 015102 MAR 2005.

[2]. Zabusky NJ, Sun Z, Peng G, "Measures of chaos and equipartition in
integrable and nonintegrable lattices"  CHAOS 16 (1): Art. No. 013130 MAR
2006.

[3] Lee DK, Peng GZ, Zabusky NJ, "Circulation rate of change: A vortex
approach for understanding accelerated inhomogeneous flows through
intermediate times."  PHYS FLUIDS 18 (9): No. 097102 SEP 2006.

[4] Peng GZ, Zabusky NJ, Zhang S, "Vortex-accelerated secondary baroclinic
vorticity deposition and late-intermediate time dynamics of a
two-dimensional Richtmyer-Meshkov interface." PHYSICS OF FLUIDS 15 (12):
3730-3744 DEC 2003.