Asymptotic behavior of monodromy. Singularly perturbed differential equations on a Riemann surface
Consider a system of first order differential equations indexed linearly by a parameter \(t\), on a compact Riemann surface \(S\): \(dm=(tA+B)m\), where \(m\) is an \(r\times r\) matrix of unknown holomorphic functions, and \(A\) and \(B\) are \(r\times r\) matrices of holomorphic one forms on \(S\). Fix initial conditions at a point \(P\in S\) and consider the function \(m(Q,t)\) defined by evaluating the solution at some other point \(Q\in S\), where the solution is obtained by analytic continuation along some path from \(P\) to \(Q\). The main result of this work is to give an asymptotic expansion for \(m(Q,t)\) as \(t\) goes to infinity in some fixed direction. Choosing \(Q=P\) and the path going from \(P\) to \(Q\) to be an element of the fundamental group of \(S\), the matrix \(m(Q,t)\) is the value of the monodromy representation of that group element. The correspondence between systems of differential equations on \(S\) and their monodromy representations is known as the Riemann-Hilbert correspondence. This correspondence is not algebraic. The parametrized family \(dm=(tA+B)m\) can be thought of as an algebraic curve in the de Rham moduli space \(M_{DR}\) of first order differential equation on \(S\). The image of this curve under the Riemann-Hilbert correspondence in the Betti moduli space \(M_ B\) of representations of the fundamental group of \(S\) is a holomorphic curve, not algebraic. Geometrically, the asymptotic expansion obtained for \(m(Q,t)\) gives a description of the behaviour of this holomorphic curve in \(M_ B\).
system of first order differential equations
scientific article; zbMATH DE number 53219