Nonlinear stability of rotating patterns (Q1003893)

The author consider \(2D\) localized rotating patterns which solve a coupled system of parabolic PDEs on the spatial domain \(\mathbb R^2\). They suggest the conditions that provide the nonlinear stability of solutions with asymptotic phase with respect to the norm in the Sobolev space \(H^2\). The stability result is obtained by a combination of energy and resolvent estimates, after the dynamics is decomposed into an evolution within a three-dimensional group orbit and a transversal evolution towards the group orbit. An application of the obtained stability result to the quintic-cubic Ginzburg-Landau equation is also discussed. In addition the relevant numerical computations are presented.