2022/12/19 by Yehorchenko, Iryna
Biochemistry, Genetics and Molecular Biology · Chemistry · Physics and Astronomy · #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Molecular spectroscopy and chirality #Nonlinear Waves and Solitons #Protein Structure and Dynamics
paper · pdf · doi:10.48550/arxiv.2212.09914
openalex publication_date 2022/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a review of some symmetry-related literature on the eikonal equations uμuμ=0,uμuμ=1, where lower indices at dependent variables designate derivatives, μ=0,1,2,..,n and summation is implied over the repeated indices. We will consider general solutions and symmetries of these equations, and relations of these equations with the reduction of higher-order PDE. Some new results that were needed for the comprehensive presentation are also adduced. In particular, we will also consider discrete symmetries of the eikonal equations equivalence classes of solutions and relations of the symmetry solutions and the general solutions. We describe in detail the procedure that allowed obtaining of the general solution using hodograph and contact transformations of the initial equations. Some new results by the author that were needed for comprehensive presentation are also adduced - in particular, a general solution of the eikonal equation for an arbitrary number of space variables. The paper is intended both as reference material and as a manual.