Diffractive patterns in a nonlinear optical two-dimensional feedback system with field rotation

Transverse Patterns in Nonlinear Optical Resonators

“…Some typical two-dimensional transformations are the rotation, translation, scaling and filtering of the pattern. The first work obtained pattern formation by controlling the spatial scale and the topology of the transverse interaction of the light field in a medium with cubic nonlinearity [220][221][222], by controlling the phase of the field with a spatial Fourier filter [223,224] (Figs. 1.9 and 1.10), and by introducing a medium with a binary-type refractive nonlinear response [225].…”

On the Asymptotics of the Solution of the Cauchy Problem for Some Differential-Difference Parabolic Equations

“…. , n. Along with (1), consider the initial condition u| t=0 = u 0 (x) ( 2 ) with a continuous bounded initial function u 0 in R n . Without loss of generality, we can assume that the vector h j coincides with the direction of the jth spatial coordinate, j = 1, .…”

“…Such equations arise, for example, in models of nonlinear optics [1][2][3] and also have a purely theoretical value as equations with nonclassical lower-order terms (e.g., see [4][5][6][7][8][9][10][11] and the bibliography therein); recall that the influence of lower-order terms in the parabolic case can be of fundamental importance [12]. We analyze the behavior of the solution of the Cauchy problem for this equation as t → ∞ and show that, unlike the classical case of differential equations (e.g., see [13,14]), for differentialdifference equations the stabilization of solutions is replaced by a more general phenomenon described by theorems on closeness of solutions.…”

Uniqueness of the solution of the cauchy problem for some differential-difference parabolic equations

“…Equations of the form (1) arise, for example, in problems of nonlinear optics (see [8][9][10]). The classical solvability of problem (1), (2) was proved and an integral representation of the solution was obtained in [11].…”