Conservation laws and other formulas for families of rays and wavefronts and for the eikonal equation

A. G. Megrabov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 22 (2019) no. 4, pp. 483-497

Résumé

In the previous studies, the author has obtained the conservation laws for the $2\mathrm{D}$ eikonal equation in an inhomogeneous isotropic medium. These laws represent the divergent identities of the form $\mathrm{div}\, F = 0$. The vector field $F$ is expressed in terms of the solution to the eikonal equation (the time field), the refractive index (the equation parameter) and their partial derivatives. Also, there were found equivalent conservation laws (divergent identities) for the families of rays and the families of wavefronts in terms of their geometric characteristics. Thus, the geometric essence (interpretation) of the above-mentioned conservation laws for the $2\mathrm{D}$ eikonal equation was discovered. In this paper, the $3\mathrm{D}$ analogs to the results obtained are presented: differential conservation laws for the $3\mathrm{D}$ eikonal equation and the conservation laws (divergent identities of the form $\mathrm{div}\, F = 0$) for the family of rays and the family of wavefronts, the vector field $F$ is expressed in terms of classical geometric characteristics of the ray curves: their Frenet basis (unit tangent vector, a principal normal and a binormal), the first curvature and the second curvature, or in terms of the classical geometric characteristics of the wavefront surfaces, i. e. their normal, principal directions, principal curvatures, the Gaussian curvature and the mean curvature. All the results have been obtained based on the vector and geometric formulas (differential conservation laws and some formulas) obtained for the families of arbitrary smooth curves, the families of arbitrary smooth surfaces and arbitrary smooth vector fields.