A New Perturbative Approach in Nonlinear Singularity Analysis

Problem statement: The study is devoted to the “mirror” method which enables one to study the integrability of nonlinear differential equations. Approach: A perturbative extension of the mirror method is introduced. Results: 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. Conclusion/Recommendations: In particular, for a nonprincipal 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.