The thesis is devoted to the "mirror" method which enables one to study the integrability of nonlinear differential equations (both ODEs and PDEs). The relevant study dates back to one century ago when Painleve made an in-depth study of singularities and initiated the (now named) Painleve analysis of integrability....[ Read more ]

The thesis is devoted to the "mirror" method which enables one to study the integrability of nonlinear differential equations (both ODEs and PDEs). The relevant study dates back to one century ago when Painleve made an in-depth study of singularities and initiated the (now named) Painleve analysis of integrability.

The mirror method uses the new tool in singularity analysis: mirror transformations and regular mirror systems, which was first introduced by Hu and Yan in 1999. In the thesis, we confine our attention to the usage of the method in several aspects. In Chapter 3, the success of constructing mirror transformations enables us to treat each principal balance in the Painleve test, singularity structures and symplectic structures of Hamiltonian systems from a common point of view. Moreover, for finite-dimensional Hamiltonian systems, the mirror transformations are canonical. In Chapter 4, the linearization of mirror systems near movable poles gives the possibility to construct the associated Backlund transformations of PDEs and the Schlesinger transformations of ODEs. Such transformations may be used to link nonlinear integrable equations to canonical forms whose properties are well known. In Chapter 5, we shall introduce a perturbative extension of the mirror method. The mirror system and its first perturbation are then utilized to gain insights into certain nonlinear equations possessing negative Fuchs indices, which were poorly understood in the literatures. In particular, for a non-principal but maximal Painleve family the first-order perturbed series solution is already a local representation of the general solution, whose convergence can also be proved.