Fluid Dynamics
Our interest in fluid dynamics is in problems with evolving free boundaries, such as shock/detonation waves and their stability, hydraulic jumps and their stability, nonlinear waves in freesurface fluid flows.
Detonation dynamics
Employing asymptotic approximations of small shock curvature and slow time variation, we develop reduced theories of detonation dynamics
Stability of a circular hydraulic jump
When a jet of fluid from your kitchen tap strikes a plate, the radial flow on the plate is initially very shallow, but at some distance away from the impact point, the depth suddenly increases, forming a ring  the hydraulic jump. Despite its apparent simplicity, the hydraulic jump is a source of many fascinating phenomena. For example, one observes that the hydraulic jump can lose stability under certain conditions, resulting in a transition from a circular to polygonal shapes.
Waves on an oscillating free suface
This topic involves the study of freesurface waves on a fluid which is being accelerated by a periodic forcing (Faraday waves) when other factors are also involved (such as rotation).
Numerical Simulations
Our primary focus is on developing highorder shockfitting methods for problems of multidimensional detonation propagation. The phenomenon of detonation poses significant challenges for numerical methods due to very complex instabilities that are inherent to detonation shocks. Very careful resolution of shock propagation is necessary to capture these instabilities. In addition, we also work on algorithms and numerical simulations of systems of reactiondiffusion equations and dispersive PDE.

Shockfitting simulation of twodimensional detonation instability
We use highorder WENO schemes combined with highorder RungeKutta method to integrate the reactive Euler equations in our shockfitting method. The movie below shows the evolution of twodimensional detonation in an ideal gas in a channel from a steady steady 1D ZND solution subject to a small initial perturbation. Due to inherent instability of the steady solution, we observe the growth of instability and subsequent formation of transverse shock waves propagating between the walls of the channel along the lead shock. The simulation algorithm is described in Taylor, Kasimov, and Stewart, Comb. Theo. Modelling (2009).

Obstaclestabilized detonation in radial outflow
A. Kasimov, S. Korneev, Detonation in supersonic radial outflow, J. Fluid Mech., 760, 313341, 2014 .

Numerical solution of dispersive PDE
With the goal of accurate numerical solution of the multidimensional nonlinear Schroedinger equation arising in the study of BoseEinstein condensate, we develop algorithms for obtaining the steadystate solutions, calculating linear stability properties, and for full timeintegration of the PDE. The movie below shows formation of vortices in a numerical simulation of the complex GrossPitaevskii equation for excitonpolariton condensate starting from a steadystate radially symmetric solution. This result is part of an MS thesis by Jesus Sierra, KAUST, 2011. For details, see J. Sierra, A. Kasimov, P. Markowich, R.M. Weishäupl, On the GrossPitaevskii equation with pumping and decay: stationary states and their stability, submitted, 2013, arXiv preprint.
Applied Partial Differential Equations
Our group's efforts are focused on the study of partial differential equations with particular emphasis on traveling wave solutions, their stability and nonlinear dynamics, asymptotic decay of stable solutions, and existence of attractors. Our primary interest is in hyperbolic PDE and shock waves.
Stability and nonlinear dynamics of shock waves
We investigate the stability of shockwave solutions in hyperbolic balance laws such as those arising in detonation theory, shallowwater flows, and continuum models of traffic (with L. Faria, R. Rosales, B. Seibold, R. Semenko, Y. Trakhinin.
Dispersive PDE
We investigate the existence of steady solutions of the GrossPitaevskii equation in the presence of sources and sinks, such as those present in modeling excitonpolariton condensates (with J. Sierra, P. Markowich, R. Weishaupl)
Decay of solutions of hyperbolic and hyperbolic/parabolic systems
We analyze the rates of decays of dissipative hyperbolic and hyperbolic/parabolic systems arizing in elastic wave propagation using energy methods (with B. SaidHouari, R. Racke)
Blowup of solutions of nonlinear wave equations
We analyze the blowup of solutions in nonlinear wave equations (with B. SaidHouari)
Reactiondiffusion systems
We investigate the existence of attractors for systems of reactiondiffusion equations arising in population biology and investigate stability of stationary solutions (with R. Parshad)