Monodromy preserving deformation of linear ordinary differential equations with rational coefficients: I. General theory and τ-function

Abstract A general theory of monodromy preserving deformation is developed for a system of linear ordinary differential equations d Y d x =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 ω, expressed rationally in terms of the coefficients of A ( x ), that has the property d ω =0 for each solution of the deformation equations. Examples corresponding to the “soliton” and “rational” solutions are discussed.

[1]  René Garnier,et al.  Sur des équations différentielles du troisième ordre dont l'intégrale générale est uniforme et sur une classe d'équations nouvelles d'ordre supérieur dont l'intégrale générale a ses points critiques fixes , 1912 .

[2]  G. Birkhoff.  The Generalized Riemann Problem for Linear Differential Equations and the Allied Problems for Linear Difference and q-Difference Equations , 1913 .

[3]  Michio Jimbo,et al.  Deformation of linear ordinary differential equations, II , 1980 .

[4]  R. Fuchs,et al.  Über lineare homogene Differentialgleichungen zweiter Ordnung mit drei im Endlichen gelegenen wesentlich singulären Stellen , 1907 .

[5]  C. Tracy,et al.  Neutron Scattering and the Correlation Functions of the Ising Model near T c , 1973 .

[6]  M. Jimbo,et al.  Aspects of Holonomic quantum fields , 1980 .

[7]  M. Ablowitz,et al.  The Inverse scattering transform fourier analysis for nonlinear problems , 1974 .

[8]  K. Ueno.  Monodromy preserving deformation and its application to soliton theory , 1980 .

[9]  A. Newell,et al.  Monodromy- and spectrum-preserving deformations I , 1980 .

[10]  Kazuo Okamoto,et al.  Polynomial Hamiltonians associated with Painlevé equations, I , 1980 .

[11]  B. McCoy,et al.  Zero-Field Susceptibility of the Two-Dimensional Ising Model near T c . , 1973 .

[12]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[13]  C. Tracy,et al.  Spin spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region , 1976 .

[14]  M. Jimbo,et al.  Holonomic Quantum Fields. II — The Riemann-Hilbert Problem , 1979 .

[15]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[16]  M. Jimbo,et al.  Holonomic Quantum Fields — The unanticipated link between deformation theory of differential equations and quantum fields — , 1980 .

[17]  Freeman J. Dyson,et al.  Fredholm determinants and inverse scattering problems , 1976 .

[18]  K. Ueno.  Monodromy preserving deformation of linear differential equations with irregular singular points , 1980 .