An asymptotic solution of the Hamilton-Jacobi equation concentrated near a surface

In construction asymptotic solutions of equations describing waves concentrated near moving lines or surfaces, special solutions (also asymptotic) of the Hamilton Jacobi equation play a central role. These solutions are real on a certain surface and are complex outside it. Solutions of such a type have been considered for the first time by V. P. Maslov. To give a mathematical description of some types waves not considered earlier, the authors return to solutions of the Hamilton Jacobi equations. For the applications that are kept in mind, it is necessary to describe thoroughly constructions, which lead to the required solution of the Hamilton-Jacobi equation in a proper form. This paper is devoted to such a description. Bibliography: 3 titles. © 2012 Springer Science+Business Media, Inc.