Nonlinear Diffusions exhibit a variety of interesting and
sometimes unexpected behaviors. I shall
give a brief overview of this broad research area and emphasize some
applications.

We consider the evolution of a tight binding wave packet propagating in a fluctuating potential. If the fluctuations stem from a stationary Markov process satisfying certain technical criteria, we show that the square amplitude of the wave packet, after diffusive rescaling, converges to a superposition of solutions of a heat equation.

Efficient algorithms of image restoration and data recovery are derived by exploring sparse approximations of the underlying solutions by redundant systems such as wavelet frames and Gabor frames. Several algorithms and numerical simulation results for image restoration, compressed sensing, and matrix completion will be presented in this
talk.

In 1911, A.E.H. Love formulated and solved equations governing the linear elastic deformation of planetary bodies due to tidal forces. In this talk, we show how modern computing capabilities reveal unstable behavior in Love's tidal model. We also extend his tidal model to include bodies of radially varying density and elastic properties.

Within the linear elastic framework, one cannot adequately explore the singular solutions. A nonlinear elastic model of the self gravitational deformation of a spherical body is posed. Analyzing spherical harmonic perturbations to this model allows us to explore the stability of the tidal problem. Solving the nonlinear elastic model requires a numerical method for a system of two nonlinear, integro-differential equations with highly nonlocal sixth integral term.