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Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I: General theory and \(\tau \)-function. (English) Zbl 1194.34167

Summary: A general theory of monodromy preserving deformation is developed for a system of linear ordinary differential equations \(dY/dx=A(x)Y\), where \(A(x)\) is a rational matrix. The non-linear deformation equations are derived and their complete integrability is proved. An explicit formula is found for a 1-form \(\omega \), expressed rationally in terms of the coefficients of \(A(x)\), that has the property \(d\omega =0\) for each solution of the deformation equations. Examples corresponding to the “soliton” and “rational” solutions are discussed.

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34A30 Linear ordinary differential equations and systems
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