Solution of a Scalar Two-Dimensional Nonlinear Problem of Diffraction on Objects of Arbitrary Shape

The aim of this study is to develop, construct, and implement methods for solving a nonlinear diffraction problem. The effect of a nonlinear medium specified by the Kerr law \({{k}^{2}}(x) = k_{1}^{2} + \alpha {{\left| {u(x)} \right|}^{2}}\) on the propagation of a wave through an object is examined. The differential and integral forms of the problem and the nonlinear integral equation are presented. The problem is solved on different bodies using different computational grids, and plots of convergence of iterative processes and graphical results are presented. Explicit and implicit methods for solving the integral equation are compared.