Miura transformations and symmetry groups of differential equations
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
The main aim of this work is to develop a generalization of the Miura transformation based on symmetry groups of differential equations. Some applications of the resulting transformations are presented. The central idea is a construction, called an HC-projection, which will generalize the well known Hopf-Cole transformation. With every differential equation there will be associated certain geometric objects which can be represented by new differential equations. These new differential equations inherit symmetries from the original differential equation. The reductions of these new differential equations using symmetry groups inherited from the original equation yield differential equations related to the original one by HC-projections. HC-projections lead to special types of Wahlquist-Estabrook prolongation and various other geometric structures. Two differential equations will be said to be related by an M-projection if they are each related to a common differential equation by HC-projections with one generating symmetry group containing the other as a subgroup. These M-projections will generalize the Miura transformation. They lead to a wider class of Wahlquist-Estabrook prolongations than do HC-projections. Construction of M-projections involves special Wahlquist-Estabrook prolongations of one of the differential equations related by the M-projection. These prolongations are characterized by their symmetry groups. A generalization of symmetries acting on nonlocal variables aids the construction of suitable prolongations, as do certain recursion operators. The construction of auto-Bäcklund transformations is significantly enhanced.