Dynamics of an Equation with a Large Spatially Distributed Control

Ilia Kashchenko

Yaroslavl State University , Russia

The dynamic properties of equations with spatially distributed parameters have been studied. Specifically, we examine the dynamic properties of the spatially distributed scalar complex quation with qubic nonlinearity. $$ dot{u}=(a-b|u|^2)u+Ke^{ivarphi}(intlimits_{-infty}^{infty}F(s)u(t,x+s)ds-u) $$ with the periodic boundary conditions $u(t,x+2pi)equiv u(t,x)$. The research technique is based on the special asymptotic method. In this context, the parameter $K$ is assumed to be sufficiently large: $K gg 1$. As result, special nonlinear families of generally parabolic evolution equations (without small or large parameters) have been constructed to determine the leading terms of the asymptotic representations of solutions to the original boundary value problem. The presence of the continual parameter in these families suggests that multistability is characteristic of principal system with a large spatially distributed control parameter.