Deformations of equations and structure in nonlinear problems of mathematical physics

Many processes in Nature can be described by using partial differential equations (PDEs). For instance, heat transfer is modelled by using the heat equation. In its simplest form, one denotes by h(x, t) the temperature at a point x and a time t. The heat equation balances the rate of change of h(x, t) with respect to the time t and the change in the rate of change with respect to the position x. This equation is an example of a linear PDE but there are many phenomena that require nonlinear equations. It was a great discovery around 1967–68 that some classes of nonlinear PDEs can be solved effectively. The Korteweg–de Vries equation (describing waves in shallow water) provides a well-studied example of exactly solvable nonlinear PDE. Such equations are very important and the range of their applicability is exceptionally wide (e.g., they describe waves in a canal, tsunami’s, propagation of light in nonlinear optics, and much more). For PDEs that encode processes in Nature, knowledge of physical conservation laws and symmetries is important for study of their properties. This interrelation between physics and mathematics leads to a beautiful and exciting area of research, to which this thesis intends to contribute. A crucial idea in this thesis is the concept of smooth deformation. Via synthesis of old and new geometric techniques we resolve Mathieu’s problem, which was a long-standing open problem in the geometry of exactly solvable nonlinear PDEs.