Asymptotic solution of Hamilton–Jacobi equation concentrated near surface

When constructing asymptotic solutions of equations describing waves concentrated near moving lines or surfaces, specific solutions (also asymptotical) of the Hamilton–Jacobi equation play a central role. These solutions are real on some surface and complex outside it. Solutions of such type were firstly considered by V. P. Maslov ([1, part 1]). To give mathematical description of some types of waves not considered earlier, the authors come back to the solutions of the Hamilton–Jacobi equations. For the applications that we keep in mind, it is necessary to describe thoroughly constructions leading to the solution of the Hamilton–Jacobi equation in the proper form. This paper is devoted to this sort of description.