Research our contributions to discovery

Nonlinear waves; PDEs with Hamiltonian structure; stochastics

interacting kinks

Domain walls play a key role in condensed matter physics, cosmology, and biological physics. Their dynamics determine bulk properties of materials undergoing phase transitions, as theorized by Nobel Laureate Lev Landau. We characterized a new class of weakly localized domain walls and formulated a new theory of their interactions, overturning decades of intuition. See also langevinkinks project.

Nonlinear Fourier analysis and transforms for ocean acoustics modeling (NONFATFOAM)

nonlinear Fourier transform

The extension of normal mode analysis to nonlinear systems is a pressing unsolved problem in engineering. The theory of nonlinear evolution equations offers an approach to generalize classical Fourier analysis on a case-by-case basis, providing a novel approach towards signal processing of nonlinear time series.

Anomalous scalings in the diffusion of granular materials

granular tumber

Check out our paradigm-shifting paper on the intermediate asymptotics of non-degenerate diffusion equations, contributed to the Proceedings of the National Academy of Sciences of the USA by G. I. Barenblatt himself. Research supported by the National Science Foundation.

Chaotic mixing of granular materials; 3D nonlinear dynamical systems and chaos

granular mixing eigenmode

This was the topic of the TMNT-Lab PI's PhD dissertation. He applied advanced techniques from the theory of nonlinear dynamical systems to the problem of mixing and segregation of granular materials. Research supported by the National Science Foundation.

Hyperbolic heat conduction and wave propagation

heat pulse evolution

Amongst a number of mathematical contributions to this field, mistakes were found in a recent paper claiming cloaking of hyperbolic heat waves. Research proudly performed on nights & weekends without expending any taxpayer funds.

Nonlinear acoustics (inviscid and thermoviscous) and shock formation

acoustic shock formation

Amongst a number of accomplishments and contributions to this field, new weakly-nonlinear acoustic equations were derived for both lossless gasses and liquids, without suffering from the shortcomings of the approximations credited to Kuznetsov and Lighthill–Westervelt (though this attribution is somewhat incorrect).

Numerical methods for hyperbolic systems of conservation laws

Legacy project, now archived.

Wavelets and wavelet methods for signal processing and solution of PDEs​

Legacy project, now archived.