×

On the isomonodromic deformation of linear ordinary differential equations of higher order. (English) Zbl 0544.34005

Let X be a Riemann surface with x a variable point and \(\omega\) a meromorphic differential 1-form, let U be a simply connected open subset of \(C^ N\) with coordinates \(t=(t^ 1,...,t^ N)\). Introducing the derivative D of a uniform function f by \(d_ xf(x,t)=Df(x,t)\cdot \omega\), the authors consider the differential equation \(D^ ny+p_ 1(x,t)D^{n-1}y+...+p_ n(x,t)y=0\) where \(p_ i\) are uniform functions on \(X\times U\). If the monodromy data of a fundamental system of solutions Y(x,t) are independent on t, then \(dY(x,t)=W(x,t)\sum A_ j(x,t)dt^ j\) with W the Wronskian and \(A_ j\) appropriate uniform functions satisfying the nonlinear system \[ \nabla^ nA_ j+p_ 1\nabla^{n-1}A_ j+...+p_ nA_ j+\partial P/\partial t^ j=0,\quad d\Omega +\Omega \wedge \Omega =0, \] where \(P=^ t(p_ n,...,p_ 1)\), \(\Omega =\sum B_ jdt^ j\), \(B_ j=(A_ j,\nabla A_ j,...,\nabla^{n-1}A_ j)\). The authors consider interrelations between monodromy data of two fundamental solutions, the case \(p_ 1\equiv 0\) and the apparent singularities.
Reviewer: J.Chrastina

MSC:

34M99 Ordinary differential equations in the complex domain
PDFBibTeX XMLCite