Bernard Deconinck's research

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The main topic underlying most of my research is the study of Nonlinear Wave Phenomena. I am especially interested in applications of Soliton Theory to Fluid Mechanics, Bose-Einstein condensates and Plasma Physics.

In the past I have used techniques from such diverse areas as Ordinary and Partial Differential Equations, Hamiltonian Systems, Asymptotics and Perturbation Theory, Lie Algebras, Riemann Surfaces and Algebraic Geometry.

Click here to know more about what you should do/know to work with me.

Some Recent Talks I've given

5. Cnoidal wave solutions of the KdV equation are linearly stable.
4. The pole dynamics of rational solutions of the Burgers equation.
3. Surface water waves: a biased overview.
2. Computing spectra of linear operators.
1. Riemann surfaces and nonlinear waves. Note: most of the fancy graphics in this talk is due to Matt Patterson.

Current Research Topics

Bose-Einstein condensates

Bose-Einstein condensates (BECs) were predicted in 1924 by Einstein as a direct consequence of Boson statistics. Experimental verification had to wait until 1995 (CU, MIT and Rice). The mean-field description of the dynamics of a BEC is given by the Nonlinear Schrodinger equation with an external potential. For some applications this potential is periodic. Depending on the application, the model is 1,2,- or 3-dimensional. We have considered both trigonometric and elliptic potentials. For all dimensions, we have constructed exact solutions. We especially look at their stability properties. The evolution of the phase of a perturbed stationary solution of the NLS equation with periodic potential.

Collaborators on this project include Jared C. Bronski, Lincoln D. Carr , Ricardo Carretero-Gonzalez , J. Nathan Kutz , Matt Patterson , Keith Promislow , and Brandon Warner .

The Kadomtsev-Petviashvili equation

My main research topic is the study of finite-genus solutions of the Kadomtsev-Petviashvili (KP) equation. The KP equation describes the evolution of long, almost one-dimensional waves in shallow water. The equation has a large class of quasi-periodic solutions which are parametrized by compact, connected Riemann surfaces of arbitrary genus. This parametrization is very transcendental. Making these solutions effective, to the point where they can be compared to experiments, is one of the focal points of my research. This is a long-term project which has led to the construction of Maple programs for black-box computation with Riemann surfaces. A genus 3 solution of the KP equation

Collaborators on this project include Mark van Hoeij , Matt Patterson and Harvey Segur .

Stability of Nonlinear Waves

Stability plays an essential role in many branches of science and engineering, including several aspects of fluid mechanics, high-speed transmission of information, and feasibility of MHD fusion devices. The objective of this project is to devise new methods for examining stability.

Thus far, we have examined different aspects of stability and instability of nonlinear waves using both numerical (see SpectrUW) and analytical techniques.

The growth rates of transverse perturbations of the one-dimensional soliton solution of the two-dimensional hyperbolic NLS equation

Collaborators on this project include John Carter , J. Nathan Kutz , David Nicholls, Dmitry Pelinovsky Harvey Segur .

Surface waves in water of arbitrary depth

This is a research project involving many. The objective of the research group is to construct the building blocks of a practical theory for inviscid surface water waves: stable or long-lived wave patterns in arbitrary depth that are fully three-dimensional and fully nonlinear. To this end, we are using methods from analysis, computational mathematics, asymptotics and algebraic geometry, together with state-of-the-art physical experiments.

Waves play an important role in the open ocean and in coastal regions. The potential coherence of three-dimensional wave patterns over large scales is not well-known, nor are the effects of coherence and large amplitude wave weather, air-sea transport processes, and large-scale structures. Thus, an understanding of these patterns is of importance to shipping and coastal engineering.

A picture of a storm caused by Hurricane Grace, the Halloween Storm of 1991.

Collaborators on this project include John Carter , Walter Craig , Joe Hammack , Diane Henderson , David Nicholls, Harvey Segur , and Catherine Sulem .

Former and current students/postdoctoral fellows

Undergraduate Students

1. Matt Patterson (1999-2001, Stability of multicomponent Bose-Einstein condensates), continued as a Ph.D. student at the University of Washington.
2. Brandon Warner (2000-2001, Stability of multicomponent Bose-Einstein condensates).
3. Mike Nivala (2003-2004, Symbolic computation of conserved quantities), continued as a Ph.D. student at the University of Washington.
4. Firat Kiyak (2004-2007, main developer of SpectrUW), graduate student in Computer Science at the University of Illinois, Urbana-Champaign.
5. Diana Widjaja (2005-2007, SpectrUW manual), graduate student in Information Technology Management, Carnegie Mellon University

Graduate Students

1. Matt Patterson (2002-2007, Computing the Abel map and the Riemann constant vector, thesis), continued as a Chowla Postdoctoral fellow at Penn State University.
2. Mike Nivala
3. Katie Oliveras
4. Chris Curtis

Postdoctoral fellows

1. Roger Thelwell (2003-2006, stability of nontrivial phase solutions of the Nonlinear Schrodinger Equation), currently Assistant Professor at James Madison University.

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