Asymptotic solution of a linear system with a high-order rotation point

An asymptotic solution of a linear system with a high order rotation point is presented. A linear system is considered in which the derivative has a small coefficient and the matrix is present in the form of A(t)+hB(t), and the asymptotics for A(t) is having three or more coinciding characteristics roots. Linearizing a nonlinear hyperbolic system of partial differential equations gives a linear hyperbolic system with three or more coinciding characteristic roots at the point of the change of multiplicity. The leading term of the asymptotics for the initial problem is obtained by applying the inverse Fourier h^-1 transforms to the components.