Abstract
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.
Highlights
The relevant literature study dates back to one century ago when Painleve made an in-depth study of singularities and initiated the Painleve analysis of integrability
Order-one reduces to a linearization of regular mirror systems, which was first introduced by mirror system near a regular singularity and allows
To demonstrate explicitly the extension of the mirror method we introduce the corresponding mirror transformation for the original equation given by (7): Bureau’s equation
Summary
The relevant literature study dates back to one century ago when Painleve made an in-depth study of singularities and initiated the ( named) Painleve analysis of integrability. In the current work our primary goal is to introduce an improvement of the mirror method so that negative indices (“resonances”) can be treated. The perturbative Painleve method is first introduced. The reason why solution close to it, represented as a perturbation the methods cannot handle negative indices lies in series in a small parameter ε. Order-zero is the usual mirror singularity analysis: mirror transformations and system. Order-one reduces to a linearization of regular mirror systems, which was first introduced by mirror system near a regular singularity and allows. Higher orders lead to the analysis of a linear, the success of constructing mirror transformations Fuchsian type inhomogeneous system. An illustrative example of Bureau’s equation structures of Hamiltonian systems from a common is presented and the conclusion follows
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