Multibump, Blow-Up, Self-Similar Solutions of the Complex Ginzburg--Landau Equation.

In this article we construct, both asymptotically and numerically, multibump, blow-up, self-similar solutions to the complex Ginzburg--Landau equation (CGL) in the limit of small dissipation. Through a careful asymptotic analysis, involving a balance of both algebraic and exponential terms, we determine the parameter range over which these solutions may exist. Most intriguingly, we determine a branch of solutions that are not perturbations of solutions to the nonlinear Schrödinger equation (NLS); moreover, they are not monotone, but they are stable. Furthermore, these axisymmetric ring-like solutions exist over a broader parameter regime than the monotone profile. [ABSTRACT FROM AUTHOR]