Geodesic
Finding exact solutions to the two-dimensional eikonal equation
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 2, pp. 201-209

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In this paper, the two-dimensional eikonal equation $f_x^2+f_y^2=\phi^2$, where $\phi=\frac1{v}$, and $v(x,y)$ is a wavespropagation velocity, is discussed. This non-linear equation is reduced to a quasilinear equation for a new dependent variable $u$. For some kinds of the functions $\phi$, solutions to the quasilinear equations are found. This means that it is possible to solve the original equation for such $\phi$. This paper also offers an approach to finding a new solution based on a known one. 
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     author = {E. D. Moskalenskii},
     title = {Finding exact solutions to the two-dimensional eikonal equation},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {201--209},
     publisher = {mathdoc},
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     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2009_12_2_a6/}
}
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E. D. Moskalenskii. Finding exact solutions to the two-dimensional eikonal equation. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 12 (2009) no. 2, pp. 201-209. http://geodesic.mathdoc.fr/item/SJVM_2009_12_2_a6/