Geodesic
Breaking of two-dimensional relativistic electron oscillations under small deviations from axial symmetry
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 11, pp. 1844-1859

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A numerical asymptotic model for the breaking of two-dimensional plane relativistic electron oscillations under a small deviation from axial symmetry is developed. The asymptotic theory makes use of the construction of time-uniformly applicable solutions to weakly nonlinear equations. A special finite-difference algorithm on staggered grids is used for numerical simulation. The numerical solutions of axially symmetric one-dimensional relativistic problems yield two-sided estimates for the breaking time. Some of the computations were performed on the “Chebyshev” supercomputer (Moscow State University). 
@article{ZVMMF_2017_57_11_a8,
     author = {A. A. Frolov and E. V. Chizhonkov},
     title = {Breaking of two-dimensional relativistic electron oscillations under small deviations from axial symmetry},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1844--1859},
     publisher = {mathdoc},
     volume = {57},
     number = {11},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_11_a8/}
}
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A. A. Frolov; E. V. Chizhonkov. Breaking of two-dimensional relativistic electron oscillations under small deviations from axial symmetry. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 11, pp. 1844-1859. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_11_a8/