Geodesic
Solution of a scalar two-dimensional nonlinear diffraction problem for objects of arbitrary shape
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 165 (2023) no. 2, pp. 167-177

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In this study, the development, design, and software implementation of the methods for solving the nonlinear diffraction problem were performed. The influence of nonlinear medium defined by the Kerr law ${k^2}\left( x \right) = k_1^2 + \alpha {\left| {u\left( x \right)} \right|^2}$ on the propagation of a wave passing through an object was examined. The differential and integral formulations of the problem and the nonlinear integral equation were considered. The problem was solved for different bodies with the use of various computational grids. Convergence graphs of the iterative processes were generated. The obtained graphical results were presented. The explicit and implicit methods for solving the integral equation were compared.
Keywords: integral equation, scalar nonlinear diffraction problem, collocation method, iterative process, numerical method. 
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     author = {A. O. Lapich and M. Yu. Medvedik},
     title = {Solution of a scalar two-dimensional nonlinear diffraction problem for objects of arbitrary shape},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {167--177},
     publisher = {mathdoc},
     volume = {165},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2023_165_2_a5/}
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A. O. Lapich; M. Yu. Medvedik. Solution of a scalar two-dimensional nonlinear diffraction problem for objects of arbitrary shape. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 165 (2023) no. 2, pp. 167-177. http://geodesic.mathdoc.fr/item/UZKU_2023_165_2_a5/