Research Highlights

Stability of Coherent Structures

Coherent Structures The kink-anti-kink profile of a bacterial colony, the local deformations of an elastic filament, or the ultra-short pulses propagating in an optical fiber, are all examples of coherent structures. Such objects are often described as special solutions of one or more model partial differential equations, and the question of their stability, as predicted by these evolution equations, arises naturally.

Modern analytical techniques have been developed to answer such questions. For instance, the Evans function is an analytic function which vanishes on the point spectrum of a linear operator. Asymptotic expansions can sometimes be used to prove that this function has a zero on the positive real axis, leading to an instability result. But often, a numerical investigation of the number of zeros of the Evans function in the right half complex plane is necessary to obtain complete spectral stability. In the case of Hamiltonian systems, other techniques are available, which lead to analytical stability results.

Such tools and methods, which combine analysis and numerical simulations, are being applied to a variety of evolution partial differential equations, relevant to problems in physics and biology.

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Dynamics of Bacterial Colonies

Bacterial Colonies Recent experiments with bacterial colonies growing on agar plates have revealed the existence of large-scale collective motions, in the form of whirls and jets. This is particularly intriguing since one has to understand how the information to create such global structures goes from the microscopic level of one bacterium to the mesoscopic level of hundreds of bacteria. These motions also influence the way the colonies develop, which occurs at a macroscopic level.

This problem falls in the general class of complex systems, in which important interactions occur at various scales, but presents the added complication of involving living organisms. Macroscopic aspects of this system can be understood in terms of a hydrodynamic model for the complex fluid made of bacteria and water. This model, which generalizes the description of bacterial colonies in terms of reaction-diffusion equations, is able to reproduce a variety of colony shapes. It also suggests that chemotactic-like behaviors in dense colonies may be a consequence of having bacteria move away from overcrowded regions. Finally, if the macroscopic contribution of the microscropic motions of bacterial flagella is described as a small-scale forcing, this hydrodynamic model is able to reproduce whirls and jets, and can be used to understand how hydrodynamic motions affect colony development.

The next step in this study is to model microscopic interactions, and use the results to formally obtain the various terms appearing in a macroscopic description of this system.

  • Experiments (made by former undergraduate student Cathy Ott, in Professor Mendelson's lab), including a movie showing large-scale bacterial motions in the form of whirls and jets
  • Hydrodynamics of bacterial colonies: model, phase diagrams and movies

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Online Research Talks

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Research Groups

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Undergraduate Research Projects

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