The main goal of the FQM-201 group is the mathematical study of several phenomena that appear on experimental sciences: Physics, Engineering, Environmental Sciences, Financial Mathematics, etc. These phenomena can be modelled in terms of systems of nonlinear partial differential equations.
During the last few years, the main topic for the research work of the group has been the application of the transformation groups theory to partial differential equations. Some specific models that have been studied are the following:
1. Some classes of equations that describe the motion of waves.
2. Some classes of equations that arise in the frame of stochastic control theory.
3. Convection diffusion equations on a porous media.
4. Lax pairs.
5. Lubrication and thin film models.
Related with these models, several symmetry reductions, equivalence transformations, differential invariants and conservation laws have been derived. By applying some direct methods, such as the expansion method, some exact solutions with physical interest have been obtained.
Moreover, the development of new methods, such as nonclassical potential symmetries, the use of nonlocal symmetries to reduce the order of some interesting ordinary differential equations that do not have Lie symmetries, the generalization of a new procedure for deriving nonclassical symmetries has been done. The connection between hidden symmetries and weak symmetries for partial differential equations has been proved. The generalization of the concept of self-adjointness that allow us to derive conservation laws for inhomogeneous equations has been introduced.
Some members of the group have also been doing a theoretical analysis on the reduction and conservation of symmetries of ordinary differential equations (ODE). They have introduced a new type of symmetry that generalizes the concept of symmetry for an ODE. The λ-symmetry concept was in part originated to investigate the conservation of Lie point symmetries in order reductions procedures. The study provided new methods to reduce the order or to integrate equations, even if they lack Lie point symmetries. The involved algorithms avoid the presence of nonlocal terms, usual in previous approaches (as hidden symmetries, exponential vector fields, nonlocal symmetries). Apart from the primary applications to reduce or integrate equations, their applicability has grown considerably since the foundation of the theory.
The main developments done by members of the group during the last years include:
1. Constructive algorithms based on the existence of λ-symmetries to calculate first integrals and integrating factors.
2. Studies on the classification of second-order equations that admit special types of λ-symmetries and algorithms to linearize and to find their analytical solutions and/or first integrals.
3. Characterizations of the second-order equations that can be linearized through both local and nonlocal transformations