Abstract:

Lie group theory is employed extensively in finding exact solutions and reductions of differential equations and their systems; it is an effective tool for finding exact solutions of nonlinear systems which arise frequently as models of various phenomena occurring in fields as diverse as physics, biology, and economics.  Based on the accuracy and insightful analysis that exact solutions of differential equations offer, they are considered a more reliable and generally more efficient alternative to the numerous numerical techniques in use.  In aiming to construct a unifying general theory of integration, Sophus Lie successfully showed that knowledge of a symmetry group of a differential equation leads to quadrature or reduction in order.    

For our research work, we intend to focus on using Lie Symmetry methods (in place of the numerical methods that have been employed in the pertinent papers) in order to find solutions and analyze the nonlinear Maccari System (MS) which successfully models 2+1 dimensional vibrations that arise in water waves as well as in phenomena occurring in plasma and atmosphere physics, nonlinear optics, Bose-Einstein condensates and superconductivity.  This system also has the rogue wave (enormous freak wave) as a solution if the group velocity along one direction vanishes.   Additional systems we would like to explore are a model of metastatic tumor evolution and a coupled PDE-ODE system which arises in phenomena such as solid-gas interaction of heat diffusion, feedback control and in the physics of power converters connected to transmission lines.  Our primary tools will be the double and multi-reduction methods and the method of optimal systems of subalgebras.    

Further plans of research include examining the dynamics of the Leaky Integrate and Fire (LIF) Oscillator with colored noise input, a nonlinear model widely employed to describe neural circuits and based on the system response of a neuron that receives fluctuating input.  We hope to arrive at a precise solution by finding the symmetries of this system of stochastic differential equations.  

Proposal Defense Committee     

1. Dr. Amer Rasheed
2. Dr. Imran Naeem (supervisor)
3. Dr. Ali Ashher Zaidi