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

Abstract

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.

Multi-bump, self-similar, blow-up solutions of the Ginzburg–Landau equation

“…The solutions found in [6] are radially symmetric and self-similar as in the case of the NLS. For the NLS, they were studied using the method of dynamical rescaling, and we also use it here.…”

“…The standard form of the GL as given in [16] can be obtained by rescaling. Numerical simulations show that there exist sets of initial data for the GL such that the solutions become infinite in finite time for 2 < d < 4, see [6,19]. Hence, a contraction of the wave packet takes place, and simultaneously the amplitude grows and blows up.…”

“…Here, we extend and adjust the techniques of [15,20,21] to prove existence and local uniqueness of the blowup solutions of the GL as found numerically in [6]. The solutions found in [6] are radially symmetric and self-similar as in the case of the NLS.…”

“…In this article we study blowup solutions to the GL as found in the numeric simulations and asymptotic analysis in [6]. We assume ε 1 such that equation (1.1) is a small perturbation of the well known nonlinear Schrödinger equation (NLS).…”

“…This method exploits the asymptotically self-similar behaviour of the solutions. Following [6], space, time, and Φ are scaled by factors of a suitably chosen norm of the solutions, denoted by L(t), which blows up at the singularity, ξ ≡ |x| L(t) , τ ≡ t 0 1 L 2 (s) ds, u(ξ, τ ) = L(t)Φ(x, t).…”