Abstract
We consider the scalar complex equation with spatially distributed control. Its dynamical properties are studied by asymptotic methods when the control coefficient is either sufficiently large or sufficiently small and the function of distribution is either almost symmetric or significantly nonsymmetric relative to zero. In all cases we reduce original equation to quasinormal form – the family of special parabolic equations, which do not contain big and small parameters, which nonlocal dynamics determines the behaviour of solutions of the original equation.
Highlights
Исследованию динамических свойств уравнений с пространственно-распределенными параметрами посвящено значительное число работ
We study dynamical properties of a complex equation with spatially-distributed parameters
The Dynamics of Kuramoto Equation with Spatially-Distributed Control
Summary
Исследованию динамических свойств уравнений с пространственно-распределенными параметрами посвящено значительное число работ (см., например, [1, 2, 3, 4, 5]). Краевая задача (1), (2) может иметь решения вида бегущих волн uk(t, x) = ρk exp(iωkt + ikx), где k целое: k = 0, ±1, ±2, . В этом случае будет показано, что для краевой задачи (1), (2) характерны быстро осциллирующие, т.е. Тогда найдется такое μ0 > 0, что при всех 0 < μ < μ0 все бегущие волны неустойчивы.
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