On asymptotic decomposition for solutions of systems of differential equations in the case of multiple roots of the characteristic equation (Q2711767)

This paper is a survey of results on the asymptotic expansion of solutions to linear \(n\)-dimensional systems of the form \(\dot x=A(\varepsilon t,\varepsilon)x\) with \(A(\tau,\varepsilon)=\sum_{s=0}^{\infty}\varepsilon^sA_s(\tau)\) and \(\varepsilon \) is a small parameter.NEWLINENEWLINENEWLINEThe author starts from a short historical review of Schlesinger-Birkhoff-Tamarkin theory and investigations of Trzitzinsky, Pugachov, Feshchenko. Next he presents his own results concerning the case when the characteristic equation \(\det(A_0(\tau)-\lambda \text{Id})=0\) has a multiple root \(\lambda_0(\tau)\) with constant structure of the corresponding root space. In this situation, under certain additional conditions, the system has a formal solution of the form NEWLINE\[NEWLINEx=\sum_{s=0}^{\infty}\mu^su_s(\tau)\exp\left(\int_{0}^{t} \sum_{s=0}^{\infty}\mu^s\lambda_s(\tau) dt\right),NEWLINE\]NEWLINE with \(\tau =\varepsilon t\), \(\mu =\varepsilon^{1/k}\), and \(k\) is the multiplicity of \(\lambda_0(\tau) \). If the structure of the root space is not constant and turning points appear, the author constructs a formal asymptotic solution under the assumption that for some integer \(k\) the roots of the equation NEWLINE\[NEWLINE\det (A_0(\tau)+ \varepsilon A_1(\tau)+\cdots +\varepsilon^kA_k(\tau)-\lambda \text{Id})=0NEWLINE\]NEWLINE are simple for all \(\tau \in[0,L]\). The author discusses some open problems.NEWLINENEWLINEFor the entire collection see [Zbl 0937.00046].