Multiperiodic solution of linear hyperbolic in the narrow sense system with constant coefficients

Abstract

There is researched existential problem of a unique multiperiodic in all independent variables solution of a linear hyperbolic in the narrow sense system of differential equations with constant coefficients and its integral representation in vector-matrix form. To solve this problem, based on Cauchy’s method of characteristics, a constructing methodology for solutions of initial problem system under consideration with various differentiation operators in vector fields directions of independent variables space has been developed based on projectors. Using this method, Cauchy problems for linear system with integral representation are solved. The introduced projectors by definition characteristic had significant value. By solving the main problem necessary and sufficient conditions for existence of multiperiodic solutions linear homogeneous systems other than trivial are established. The conditions are obtained for absence of nonzero multiperiodic solutions of these systems. In absence of nonzero multiperiodic solutions linear homogeneous systems, the main theorem on existence and uniqueness of multiperiodic solution linear nonhomogeneous system with derivation of its integral representation depending on projection operators is proved. The developed method has prospect of extending the results to quasilinear system under consideration, as well as to multidimensional vector t = (t1; :::; tm) and multiperiodic matrices at partial derivatives of unknown vectorfunction.