The Jacobi–Maupertuis principle and Fermat variational principle in the problem of short-wave asymptotics in the solution of the Helmholtz equation with a localized source

The problem of short-wave asymptotics of the Helmholtz equation with a localized right-hand side in the form of a rapidly decreasing function is considered in the article. We present an algorithm for calculating rays using the variational method and the wave field applying the canonical Maslov operator method for given boundary conditions. This approach is used for model examples, including those with a logarithmic feature of the ray family. In addition, we consider applications of the variational method for calculating rays in the illuminated region and in the caustic shadow region.