Abstract

For a problem arising in nonlinear optics, namely, for an initial-boundary value problem in a disk for a nonlinear parabolic equation with time delay and rotation of spatial argument by a given angle, bifurcations of self-oscillatory solutions from homogeneous equilibrium states are studied. In the plane of basic parameters of the equation, domains of stability (instability) of homogeneous equilibrium states are constructed, and the dynamics of the stability domains is analyzed depending on the delay value. The mechanisms of stability loss of homogeneous equilibrium states are investigated, possible bifurcations of spatially inhomogeneous self-oscillatory solutions and their stability are analyzed, and the dynamics of such solutions near the boundary of a stability domain in the plane of basic parameters of the equation is studied.

中文翻译：

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具有旋转空间参数和时滞的非线性抛物方程的自振动解的分歧

摘要

对于非线性光学中出现的问题，即对于具有时间延迟和空间参数旋转给定角度的非线性抛物线方程的圆盘中的初始边界值问题，研究了自齐次平衡态自振荡解的分歧。在方程的基本参数平面上，构造了均匀平衡状态的稳定性（不稳定性）域，并根据延迟值分析了稳定性域的动力学。研究了均质平衡态稳定性损失的机理，分析了空间非均匀自振荡解的可能分支以及它们的稳定性，并且在等式基本参数平面内稳定域边界附近的此类解的动力学为：研究过。