Asymptotic integration of a Cauchy problem (Q1063781)

The aim of the paper is to justify the construction of an asymptotic solution of the Cauchy problem: \[ (1)\quad \underline u_ t+{\mathcal D}\underline u_ x=\epsilon A\underline u,\quad u(x,0)=\sum \epsilon^ j\psi_ j(x),\quad (t,x)\in [0,T]\times]-\infty,+\infty [, \] where \({\mathcal D}\) is a diagonal matrix \((\lambda_ 1,...,\lambda_ n)\) and A is an (n\(\times n)\) matrix. The solution is sought to be valid in a band \(T=O(\epsilon^{-2})\). First, a solution is set in the form: \[ u_ i=v_{i0}(y_ i,\tau,\xi)+\sum_{k>0}\epsilon^ k(v_{ik}(y_ i,\tau,\xi)+w_{ik}(y_ i,\tau,\xi,t)) \] where \(y_ i=x-\lambda_ it\), \(\tau =\epsilon t\), \(\xi =\epsilon^ 2t\). The \(v_{ik}\) and \(w_{ik}\) are formally determined by identifying the coefficients of identical powers of \(\epsilon\). The slow variation of the \(v_{ik}\) is determined by using orthogonality conditions. As in many Russian papers, the only literature cited is Soviet literature. In particular, J. B. Keller's papers are not mentioned.