Grant: Discovery Grants Program - Individual

The proposed research concentrates on analytical methods for solving systems of nonlinear PDEs which appear in various branches of Mathematical Physics. The subject of this investigation involves symmetry reduction methods and geometric methods based on the analysis of surfaces immersed in Lie algebras. Certain applications to the equations of nonlinear field theory and fluid dynamics will be studied. The development of these methods provides us with new tools for dealing with nonlinear phenomena in physics, especially the ones which are described by multidimensional systems of PDEs and which so far have not been approached successfully by other methods. This program includes the following projects: 1. Surfaces in Lie Algebras and Homogeneous Spaces via Grassmannian Models This project employs a new way of constructing and studying properties of surfaces immersed in multidimensional spaces through a combination of the techniques of group-theoretical analysis and completely integrable sigma models. 2. Variational Extension of Invariant Solutions of Evolution Systems This project aims at finding new classes of physically interesting approximate solutions of nonlinear evolution equations by using variational methods which introduce time variation parameters to group invariant solutions. 3. Invariant Solutions of Fluid Dynamics Eqs. via the Weak Transversality Method This project involves a systematic construction of the invariant and partially invariant solutions of the Navier-Stokes and mhd eqs. in (3+1) dimensions, obtained by weakening the strong transversality condition required by the symmetry reduction method. 4. Rank-k Solutions for Multidimensional Hydrodynamic-Type Systems A new version of the conditional symmetry method for constructing rank-k solutions expressible in terms of Riemann invariants is proposed for hydrodynamic-type systems.